Theorem axsep 2775

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 2766. 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 1983. 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 2778, 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 2814 shows (contradicting zfauscl 2778). However, as axsep2 2777 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 2776 from ax-rep 2766.

This theorem should not be referenced by any proof. Instead, use ax-sep 2776 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

Step	Hyp	Ref	Expression
1		ax-17 1006	. . . 4 |- ((w = x /\ ph) -> A.y(w = x /\ ph))
2	1	axrep5 2771	. . 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 1160	. . . . 5 |- E.y y = w
4		equtr 1167	. . . . . . . . 9 |- (y = w -> (w = x -> y = x))
5		equcomi 1164	. . . . . . . . 9 |- (y = x -> x = y)
6	4, 5	syl6 22	. . . . . . . 8 |- (y = w -> (w = x -> x = y))
7	6	adantrd 391	. . . . . . 7 |- (y = w -> ((w = x /\ ph) -> x = y))
8	7	19.21aiv 1323	. . . . . 6 |- (y = w -> A.x((w = x /\ ph) -> x = y))
9	8	19.22i 1075	. . . . 5 |- (E.y y = w -> E.yA.x((w = x /\ ph) -> x = y))
10	3, 9	ax-mp 7	. . . 4 |- E.yA.x((w = x /\ ph) -> x = y)
11	10	a1i 8	. . 3 |- (w e. z -> E.yA.x((w = x /\ ph) -> x = y))
12	2, 11	mpg 1021	. 2 |- E.yA.x(x e. y <-> E.w(w e. z /\ (w = x /\ ph)))
13		an12 486	. . . . . . 7 |- ((w = x /\ (w e. z /\ ph)) <-> (w e. z /\ (w = x /\ ph)))
14	13	exbii 1086	. . . . . 6 |- (E.w(w = x /\ (w e. z /\ ph)) <-> E.w(w e. z /\ (w = x /\ ph)))
15		ax-17 1006	. . . . . . 7 |- ((x e. z /\ ph) -> A.w(x e. z /\ ph))
16		elequ1 1172	. . . . . . . 8 |- (w = x -> (w e. z <-> x e. z))
17	16	anbi1d 619	. . . . . . 7 |- (w = x -> ((w e. z /\ ph) <-> (x e. z /\ ph)))
18	15, 17	equsex 1188	. . . . . 6 |- (E.w(w = x /\ (w e. z /\ ph)) <-> (x e. z /\ ph))
19	14, 18	bitr3i 173	. . . . 5 |- (E.w(w e. z /\ (w = x /\ ph)) <-> (x e. z /\ ph))
20	19	bibi2i 610	. . . 4 |- ((x e. y <-> E.w(w e. z /\ (w = x /\ ph))) <-> (x e. y <-> (x e. z /\ ph)))
21	20	albii 1034	. . 3 |- (A.x(x e. y <-> E.w(w e. z /\ (w = x /\ ph))) <-> A.x(x e. y <-> (x e. z /\ ph)))
22	21	exbii 1086	. 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)))
23	12, 22	mpbi 187	1 |- E.yA.x(x e. y <-> (x e. z /\ ph))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 989   = wceq 991   e. wcel 993  E.wex 1015

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 997  ax-gen 998  ax-8 999  ax-9 1000  ax-12 1003  ax-13 1004  ax-14 1005  ax-17 1006  ax-4 1008  ax-5o 1010  ax-6o 1013  ax-9o 1158  ax-rep 2766

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1016

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