Theorem axsep 4780

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 4771. 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 3434. 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 4783, 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 4840 shows (contradicting zfauscl 4783). However, as axsep2 4782 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 4781 from ax-rep 4771.

This theorem should not be referenced by any proof. Instead, use ax-sep 4781 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 1843	. . . 4 |- F/ y ( w = x /\ ph )
2	1	axrep5 4776	. . 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 1948	. . . . . . . 8 |- ( y = w -> ( w = x -> y = x ) )
4		equcomi 1944	. . . . . . . 8 |- ( y = x -> x = y )
5	3, 4	syl6 35	. . . . . . 7 |- ( y = w -> ( w = x -> x = y ) )
6	5	adantrd 484	. . . . . 6 |- ( y = w -> ( ( w = x /\ ph ) -> x = y ) )
7	6	alrimiv 1855	. . . . 5 |- ( y = w -> A. x ( ( w = x /\ ph ) -> x = y ) )
8	7	a1d 25	. . . 4 |- ( y = w -> ( w e. z -> A. x ( ( w = x /\ ph ) -> x = y ) ) )
9	8	spimev 2259	. . 3 |- ( w e. z -> E. y A. x ( ( w = x /\ ph ) -> x = y ) )
10	2, 9	mpg 1724	. 2 |- E. y A. x ( x e. y <-> E. w ( w e. z /\ ( w = x /\ ph ) ) )
11		an12 838	. . . . . . 7 |- ( ( w = x /\ ( w e. z /\ ph ) ) <-> ( w e. z /\ ( w = x /\ ph ) ) )
12	11	exbii 1774	. . . . . 6 |- ( E. w ( w = x /\ ( w e. z /\ ph ) ) <-> E. w ( w e. z /\ ( w = x /\ ph ) ) )
13		elequ1 1997	. . . . . . . 8 |- ( w = x -> ( w e. z <-> x e. z ) )
14	13	anbi1d 741	. . . . . . 7 |- ( w = x -> ( ( w e. z /\ ph ) <-> ( x e. z /\ ph ) ) )
15	14	equsexvw 1932	. . . . . 6 |- ( E. w ( w = x /\ ( w e. z /\ ph ) ) <-> ( x e. z /\ ph ) )
16	12, 15	bitr3i 266	. . . . 5 |- ( E. w ( w e. z /\ ( w = x /\ ph ) ) <-> ( x e. z /\ ph ) )
17	16	bibi2i 327	. . . 4 |- ( ( x e. y <-> E. w ( w e. z /\ ( w = x /\ ph ) ) ) <-> ( x e. y <-> ( x e. z /\ ph ) ) )
18	17	albii 1747	. . 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 1774	. 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 220	1 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-rep 4771

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)