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**Statement List for Metamath Proof Explorer -** 4501-4600 - Page 46 of 107
Type	Label	Description
Statement

Theorem	limenpsi 4501	A limit ordinal is equinumerous to a proper subset of itself.


Theorem	limensuci 4502	A limit ordinal is equinumerous to its successor.


Theorem	limensuc 4503	A limit ordinal is equinumerous to its successor.
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Pigeonhole Principle

Theorem	phplem1 4504	Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element.

Theorem	phplem2 4505	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.

Theorem	phplem3 4506	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.

Theorem	phplem4 4507	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.

Theorem	nneneq 4508	Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136.
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Theorem	php 4509	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 4504 through phplem4 4507, nneneq 4508, and this final piece of the proof.
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Theorem	php2 4510	Corollary of Pigeonhole Principle.
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Theorem	php3 4511	Corollary of Pigeonhole Principle. If is finite and is a proper subset of , the is strictly less numerous than . Stronger version of Corollary 6C of [Enderton] p. 135. (The expression is the definition of "finite," and "infinite" is defined as "not finite.")
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Theorem	php4 4512	Corollary of the Pigeonhole Principle php 4509: a natural number is strictly dominated by its successor.


Theorem	php5 4513	Corollary of the Pigeonhole Principle php 4509: a natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90.


Finite sets

Theorem	onomeneq 4514	An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse.
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Theorem	onfin 4515	An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92.


Theorem	nndomo 4516	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146.
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Theorem	nnsdomo 4517	Cardinal ordering agrees with natural number ordering.
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Theorem	omsucdom 4518	Strict dominance of natural numbers is the same as dominance over the successor of the smaller.
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Theorem	sucdomi 4519	Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 4832.
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Theorem	0sdom1dom 4520	Strict dominance over zero is the same as dominance over one.


Theorem	1sdom2 4521	Ordinal 1 is strictly dominated by ordinal 2.


Theorem	finsucdom 4522	Strict dominance of a finite set over a natural number is the same as dominance over its successor.
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Theorem	pssinf 4523	A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136.
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Theorem	ominf 4524	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.


Theorem	omsdomnn 4525	Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. Here we use $/\$ instead of because, due to a peculiarity ultimately caused our ordered pair definition, we would need the Axiom of infinity (which we have avoided up to now) in order to prove the latter.
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Theorem	isfinite1 4526	Omega strictly dominates a finite set. See comment in omsdomnn 4525.
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Theorem	infsdomnn 4527	An infinite set strictly dominates a natural number.
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Theorem	infn0 4528	An infinite set is not empty.


Theorem	pssnn 4529	A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137.
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Theorem	ssnn 4530	A subset of a natural number is finite.
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Theorem	ssfi 4531	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
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Theorem	domfi 4532	A set dominated by a finite set is finite.
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Theorem	unblem1 4533	Lemma for unbnn 4537. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set.

Theorem	unblem2 4534	Lemma for unbnn 4537. The value of the function belongs to the unbounded set of natural numbers .

Theorem	unblem3 4535	Lemma for unbnn 4537. The value of the function is less than its value at a successor.

Theorem	unblem4 4536	Lemma for unbnn 4537. The function maps the set of natural numbers one-to-one to the set of unbounded natural numbers .

Theorem	unbnn 4537	Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnnt 4629 for a stronger version without the hypothesis.
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Theorem	unbnn2 4538	Version of unbnn 4537 that does not require a strict upper bound.
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Theorem	isfinite2 4539	Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity.


Theorem	unfilem1 4540	Lemma for proving that the union of two finite sets is finite.

Theorem	unfilem2 4541	Lemma for proving that the union of two finite sets is finite.

Theorem	unfilem3 4542	Lemma for proving that the union of two finite sets is finite.

Theorem	fin2inf 4543	This theorem, which was proved without the Axiom of Infinity, is an artifact of our definition of strict dominance, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless exists.


Theorem	unfi 4544	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144.
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Theorem	unfi2 4545	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 4544 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4543).
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Theorem	infcntss 4546	Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.)
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Theorem	prfi 4547	An unordered pair is finite.


Theorem	unifi 4548	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144.
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Theorem	unifi2 4549	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 4548 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4543).
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Theorem	fiint 4550	Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of is in ." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally.
$/\$ $/\$

Theorem	abfii1 4551	Two ways to express the collection of finite intersections of a set .
$/\$ $/\$ $/\$ $/\$

Theorem	abfii2 4552	Two ways to express the collection of finite intersections of a set .
$/\$ $/\$ $/\$ $/\$

Theorem	abfii3 4553	Two ways to express the collection of finite intersections of a set .
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	abfii4 4554	Two ways to express the collection of finite intersections of a set . Even though the expressions differ by only one symbol, the proof is not simple.
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	abfii5 4555	Two ways to express the collection of finite intersections of a set .
$/\$ $/\$ $/\$

Theorem	fodomfi 4556	An onto function implies dominance of domain over range, for finite sets. Unlike fodom 4788 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof.
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Theorem	fodomfib 4557	Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 4790 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof.
$/\$ $/\$

Theorem	fofi 4558	If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104.
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Theorem	iunfi 4559	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 4548. Note that depends on , i.e. can be thought of as .
$/\$

Theorem	pwfilem 4560	Lemma for pwfi 4561.

Theorem	pwfi 4561	The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105.


Theorem	pm54.43 4562	Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2. Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 4823), so that their means, in our notation, i.e. (by elab 1893) i.e. (by carden 4821 and cardnn 4814). We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.) Theorem pm110.643 4913 shows the derivation of 1+1=2 for cardinal numbers from this theorem.
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Supremum

Syntax	csup 4563	Extend class notation to include supremum of class . Here is ordinarily a relation that strictly orders class . For example, could be 'less than' and could be the set of real numbers.


Definition	df-sup 4564	Define the supremum of class . It is meaningful when is a relation that strictly orders and when the supremum exists. For example, could be 'less than', could be the set of real numbers, and could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrval 6619. We will also use this notation for "infimum" by replacing with .