Theorem axsep 4540

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 4531. 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 3278. 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 4543, which requires the Axiom of Extensionality as well as Separation for its derivation.

If we omit the requirement that y not occur in ph, we can derive a contradiction, as notzfaus 4595 shows (contradicting zfauscl 4543). However, as axsep2 4542 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 4541 from ax-rep 4531.

This theorem should not be referenced by any proof. Instead, use ax-sep 4541 below so that the uses of the Axiom of Separation can be more easily identified. (Contributed by NM, 11-Sep-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
axsep	|- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )

Distinct variable groups:   x, y, z   ph, y, z

Allowed substitution hint:   ph( x)

Proof of Theorem axsep

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1772	. . . 4 |- F/ y ( w = x /\ ph )
2	1	axrep5 4536	. . 3 |- ( A. w ( w e. z -> E. y A. x ( ( w = x /\ ph ) -> x = y ) ) -> E. y A. x ( x e. y <-> E. w ( w e. z /\ ( w = x /\ ph ) ) ) )
3		equtr 1876	. . . . . . . 8 |- ( y = w -> ( w = x -> y = x ) )
4		equcomi 1872	. . . . . . . 8 |- ( y = x -> x = y )
5	3, 4	syl6 34	. . . . . . 7 |- ( y = w -> ( w = x -> x = y ) )
6	5	adantrd 474	. . . . . 6 |- ( y = w -> ( ( w = x /\ ph ) -> x = y ) )
7	6	alrimiv 1784	. . . . 5 |- ( y = w -> A. x ( ( w = x /\ ph ) -> x = y ) )
8	7	a1d 26	. . . 4 |- ( y = w -> ( w e. z -> A. x ( ( w = x /\ ph ) -> x = y ) ) )
9	8	spimev 2114	. . 3 |- ( w e. z -> E. y A. x ( ( w = x /\ ph ) -> x = y ) )
10	2, 9	mpg 1682	. 2 |- E. y A. x ( x e. y <-> E. w ( w e. z /\ ( w = x /\ ph ) ) )
11		an12 811	. . . . . . 7 |- ( ( w = x /\ ( w e. z /\ ph ) ) <-> ( w e. z /\ ( w = x /\ ph ) ) )
12	11	exbii 1729	. . . . . 6 |- ( E. w ( w = x /\ ( w e. z /\ ph ) ) <-> E. w ( w e. z /\ ( w = x /\ ph ) ) )
13		elequ1 1905	. . . . . . . 8 |- ( w = x -> ( w e. z <-> x e. z ) )
14	13	anbi1d 716	. . . . . . 7 |- ( w = x -> ( ( w e. z /\ ph ) <-> ( x e. z /\ ph ) ) )
15	14	equsexvw 1859	. . . . . 6 |- ( E. w ( w = x /\ ( w e. z /\ ph ) ) <-> ( x e. z /\ ph ) )
16	12, 15	bitr3i 259	. . . . 5 |- ( E. w ( w e. z /\ ( w = x /\ ph ) ) <-> ( x e. z /\ ph ) )
17	16	bibi2i 319	. . . 4 |- ( ( x e. y <-> E. w ( w e. z /\ ( w = x /\ ph ) ) ) <-> ( x e. y <-> ( x e. z /\ ph ) ) )
18	17	albii 1702	. . 3 |- ( A. x ( x e. y <-> E. w ( w e. z /\ ( w = x /\ ph ) ) ) <-> A. x ( x e. y <-> ( x e. z /\ ph ) ) )
19	18	exbii 1729	. 2 |- ( E. y A. x ( x e. y <-> E. w ( w e. z /\ ( w = x /\ ph ) ) ) <-> E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) )
20	10, 19	mpbi 213	1 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1453   E.wex 1674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-rep 4531

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679

This theorem is referenced by: (None)