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**Statement List for Metamath Proof Explorer -** 1401-1500 - Page 15 of 107
Type	Label	Description
Statement

Theorem	mo4 1401	"At most one" expressed using implicit substitution.
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Theorem	mobid 1402	Formula-building rule for "at most one" quantifier (deduction rule).


Theorem	mobii 1403	Formula-building rule for "at most one" quantifier (inference rule).


Theorem	hbmo1 1404	Bound-variable hypothesis builder for "at most one."


Theorem	hbmo 1405	Bound-variable hypothesis builder for "at most one."


Theorem	cbvmo 1406	Rule used to change bound variables with implicit substitution.


Theorem	eu5 1407	Uniqueness in terms of "at most one."
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Theorem	eu4 1408	Uniqueness using implicit substitution.
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Theorem	eumo 1409	Existential uniqueness implies "at most one."


Theorem	eumoi 1410	"At most one" inferred from existential uniqueness.


Theorem	exmoeu 1411	Existence in terms of "at most one" and uniqueness.


Theorem	exmoeu2 1412	Existence implies "at most one" is equivalent to uniqueness.


Theorem	moabs 1413	Absorption of existence condition by "at most one."


Theorem	exmo 1414	Something exists or at most one exists.
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Theorem	immo 1415	"At most one" is preserved through implication (notice wff reversal).


Theorem	immoi 1416	"At most one" is preserved through implication (notice wff reversal).


Theorem	moimv 1417	Move antecedent outside of "at most one."


Theorem	euimmo 1418	Uniqueness implies "at most one" through implication.


Theorem	euim 1419	Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent.
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Theorem	moan 1420	"At most one" is still the case when a conjunct is added.
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Theorem	moani 1421	"At most one" is still true when a conjunct is added.
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Theorem	moor 1422	"At most one" is still the case when a disjunct is removed.
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Theorem	mooran1 1423	"At most one" imports disjunction to conjunction.
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Theorem	mooran2 1424	"At most one" exports disjunction to conjunction.
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Theorem	moanim 1425	Introduction of a conjunct into "at most one" quantifier.
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Theorem	euan 1426	Introduction of a conjunct into uniqueness quantifier.
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Theorem	moanimv 1427	Introduction of a conjunct into "at most one" quantifier.
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Theorem	moaneu 1428	Nested "at most one" and uniqueness quantifiers.
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Theorem	moanmo 1429	Nested "at most one" quantifiers.
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Theorem	euanv 1430	Introduction of a conjunct into uniqueness quantifier.
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Theorem	mopick 1431	"At most one" picks a variable value, eliminating an existential quantifier.
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Theorem	eupick 1432	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing such that is true, and there is also an (actually the same one) such that and are both true, then implies regardless of . This theorem, which apparently does not appear explicitly in the literature, can be quite useful because it lets us eliminate existential quantifiers in a hypothesis.
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Theorem	eupickb 1433	Existential uniqueness "pick" showing wff equivalence.
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Theorem	mopick2 1434	"At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1092.
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Theorem	euor2 1435	Introduce or eliminate a disjunct in a uniqueness quantifier.
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Theorem	moexex 1436	"At most one" double quantification.
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Theorem	moexexv 1437	"At most one" double quantification.
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Theorem	2moex 1438	Double quantification with "at most one."


Theorem	2euex 1439	Double quantification with existential uniqueness.


Theorem	2eumo 1440	Double quantification with existential uniqueness and "at most one."


Theorem	2eu2ex 1441	Double existential uniqueness.


Theorem	2moswap 1442	A condition allowing swap of "at most one" and existential quantifiers.


Theorem	2euswap 1443	A condition allowing swap of uniqueness and existential quantifiers.


Theorem	2exeu 1444	Double existential uniqueness implies double uniqueness quantification.
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Theorem	2mo 1445	Two equivalent expressions for double "at most one."
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Theorem	2mos 1446	Double "exists at most one" with implicit substitution.
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Theorem	2eu1 1447	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one.
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Theorem	2eu2 1448	Double existential uniqueness.


Theorem	2eu3 1449	Double existential uniqueness.
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Theorem	2eu4 1450	This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2eu1 1447 for a condition under which the naive definition holds and 2exeu 1444 for a one-way implication. See 2eu5 1451 and 2eu8 1454 for alternate definitions.
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Theorem	2eu5 1451	An alternate definition of double existential uniqueness (see 2eu4 1450). A mistake sometimes made in the literature is to use to mean "exactly one and exactly one ." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining as an additional condition. The correct definition apparently has never been published. ( means "exists at most one.")
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Theorem	2eu6 1452	Two equivalent expressions for double existential uniqueness.
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Theorem	2eu7 1453	Two equivalent expressions for double existential uniqueness.
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Theorem	2eu8 1454	Two equivalent expressions for double existential uniqueness. Curiously, we can put on either of the internal conjuncts but not both. We can also commute using 2eu7 1453.
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Theorem	exists1 1455	Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 2768.


Theorem	exists2 1456	A condition implying that at least two things exist.
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ZF Set Theory - start with the Axiom of Extensionality

Introduce the Axiom of Extensionality

Axiom	ax-ext 1457	Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461. Set theory can also be formulated with a single primitive predicate on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes , and equality is defined as . All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8. To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 962 through ax-16 1208 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic. General remarks: Our set theory axioms are presented using defined connectives (, , etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives , , , , and . It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so. It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable in ax-ext 1457 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both and . This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 2689, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the the infinite axioms generated by the ax-ext 1457 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version.


Theorem	axext 1458	The Axiom of Extensionality (ax-ext 1457) restated so that it postulates the existence of a set given two arbitrary sets and . This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets.


Theorem	zfext2 1459	A generalization of the Axiom of Extensionality in which and need not be distinct.
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