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Theorem axsep 2670
Description: Separation Scheme, which is an axiom scheme of Zermelo's original theory. Scheme Sep of [BellMachover] p. 463. As we show here, it is redundant if we assume Replacement in the form of ax-rep 2661. Some textbooks present Separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger Replacement. The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with x e. z) so that it asserts the existence of a collection only if it is smaller than some other collection z that already exists. This prevents Russell's paradox ru 1909. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

The variable x can appear free in the wff ph, which in textbooks is often written ph(x). To specify this in the Metamath language, we omit the distinct variable requirement ($d) that x not appear in ph.

For a version using a class variable, see zfauscl 2673, which requires the Axiom of Extensionality as well as Replacement for its derivation.

If we omit the requirement that y not occur in ph, we can derive a contradiction, as notzfaus 2709 shows (contradicting zfauscl 2673). However, as axsep2 2672 shows, we can eliminate the restriction that z not occur in ph.

Note: the distinct variable restriction that z not occur in ph is actually redundant in this particular proof, but we keep it since its purpose is to demonstrate the derivation of the exact ax-sep 2671 from ax-rep 2661.

This theorem should not be referenced by any proof. Instead, use ax-sep 2671 below so that the uses of the Axiom of Separation can be more easily identified.

Assertion
Ref Expression
axsep |- E.yA.x(x e. y <-> (x e. z /\ ph))
Distinct variable groups:   x,y,z   ph,y,z
Proof of Theorem axsep
StepHypRef Expression
1 ax-17 1190 . . . 4 |- ((w = x /\ ph) -> A.y(w = x /\ ph))
21axrep5 2666 . . 3 |- (A.w(w e. z -> E.yA.x((w = x /\ ph) -> x = y)) -> E.yA.x(x e. y <-> E.w(w e. z /\ (w = x /\ ph))))
3 a9e 1112 . . . . 5 |- E.y y = w
4 equtr 1118 . . . . . . . . 9 |- (y = w -> (w = x -> y = x))
5 equcomi 1115 . . . . . . . . 9 |- (y = x -> x = y)
64, 5syl6 22 . . . . . . . 8 |- (y = w -> (w = x -> x = y))
76adantrd 391 . . . . . . 7 |- (y = w -> ((w = x /\ ph) -> x = y))
8719.21aiv 1268 . . . . . 6 |- (y = w -> A.x((w = x /\ ph) -> x = y))
9819.22i 1016 . . . . 5 |- (E.y y = w -> E.yA.x((w = x /\ ph) -> x = y))
103, 9ax-mp 7 . . . 4 |- E.yA.x((w = x /\ ph) -> x = y)
1110a1i 8 . . 3 |- (w e. z -> E.yA.x((w = x /\ ph) -> x = y))
122, 11mpg 962 . 2 |- E.yA.x(x e. y <-> E.w(w e. z /\ (w = x /\ ph)))
13 an12 483 . . . . . . 7 |- ((w = x /\ (w e. z /\ ph)) <-> (w e. z /\ (w = x /\ ph)))
1413exbii 1027 . . . . . 6 |- (E.w(w = x /\ (w e. z /\ ph)) <-> E.w(w e. z /\ (w = x /\ ph)))
15 ax-17 1190 . . . . . . 7 |- ((x e. z /\ ph) -> A.w(x e. z /\ ph))
16 elequ1 1123 . . . . . . . 8 |- (w = x -> (w e. z <-> x e. z))
1716anbi1d 615 . . . . . . 7 |- (w = x -> ((w e. z /\ ph) <-> (x e. z /\ ph)))
1815, 17equsex 1135 . . . . . 6 |- (E.w(w = x /\ (w e. z /\ ph)) <-> (x e. z /\ ph))
1914, 18bitr3 175 . . . . 5 |- (E.w(w e. z /\ (w = x /\ ph)) <-> (x e. z /\ ph))
2019bibi2i 606 . . . 4 |- ((x e. y <-> E.w(w e. z /\ (w = x /\ ph))) <-> (x e. y <-> (x e. z /\ ph)))
2120albii 975 . . 3 |- (A.x(x e. y <-> E.w(w e. z /\ (w = x /\ ph))) <-> A.x(x e. y <-> (x e. z /\ ph)))
2221exbii 1027 . 2 |- (E.yA.x(x e. y <-> E.w(w e. z /\ (w = x /\ ph))) <-> E.yA.x(x e. y <-> (x e. z /\ ph)))
2312, 22mpbi 189 1 |- E.yA.x(x e. y <-> (x e. z /\ ph))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-12 1104  ax-13 1107  ax-14 1108  ax-17 1190  ax-rep 2661
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957
Copyright terms: Public domain