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Theorem axsep 4156
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 4147. 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 3003. 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 4159, 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 4201 shows (contradicting zfauscl 4159). However, as axsep2 4158 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 4157 from ax-rep 4147.

This theorem should not be referenced by any proof. Instead, use ax-sep 4157 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.
StepHypRef Expression
1 nfv 1609 . . . 4  |-  F/ y ( w  =  x  /\  ph )
21axrep5 4152 . . 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 1671 . . . . . . . 8  |-  ( y  =  w  ->  (
w  =  x  -> 
y  =  x ) )
4 equcomi 1664 . . . . . . . 8  |-  ( y  =  x  ->  x  =  y )
53, 4syl6 29 . . . . . . 7  |-  ( y  =  w  ->  (
w  =  x  ->  x  =  y )
)
65adantrd 454 . . . . . 6  |-  ( y  =  w  ->  (
( w  =  x  /\  ph )  ->  x  =  y )
)
76alrimiv 1621 . . . . 5  |-  ( y  =  w  ->  A. x
( ( w  =  x  /\  ph )  ->  x  =  y ) )
87a1d 22 . . . 4  |-  ( y  =  w  ->  (
w  e.  z  ->  A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) ) )
98spimev 1952 . . 3  |-  ( w  e.  z  ->  E. y A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) )
102, 9mpg 1538 . 2  |-  E. y A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
11 an12 772 . . . . . . 7  |-  ( ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
1211exbii 1572 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )
13 nfv 1609 . . . . . . 7  |-  F/ w
( x  e.  z  /\  ph )
14 elequ1 1699 . . . . . . . 8  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
1514anbi1d 685 . . . . . . 7  |-  ( w  =  x  ->  (
( w  e.  z  /\  ph )  <->  ( x  e.  z  /\  ph )
) )
1613, 15equsex 1915 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1712, 16bitr3i 242 . . . . 5  |-  ( E. w ( w  e.  z  /\  ( w  =  x  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1817bibi2i 304 . . . 4  |-  ( ( x  e.  y  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )  <->  ( x  e.  y  <->  ( x  e.  z  /\  ph )
) )
1918albii 1556 . . 3  |-  ( A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )  <->  A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) ) )
2019exbii 1572 . 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 )
) )
2110, 20mpbi 199 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-cleq 2289  df-clel 2292
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