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Theorem axsep 4143
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 4134. 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 2993. 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 4146, 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 4186 shows (contradicting zfauscl 4146). However, as axsep2 4145 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 4144 from ax-rep 4134.

This theorem should not be referenced by any proof. Instead, use ax-sep 4144 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
Dummy variable  w is distinct from all other variables.
Allowed substitution group:    ph( x)

Proof of Theorem axsep
StepHypRef Expression
1 nfv 1607 . . . 4  |-  F/ y ( w  =  x  /\  ph )
21axrep5 4139 . . 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 1655 . . . . . . . 8  |-  ( y  =  w  ->  (
w  =  x  -> 
y  =  x ) )
4 equcomi 1649 . . . . . . . 8  |-  ( y  =  x  ->  x  =  y )
53, 4syl6 31 . . . . . . 7  |-  ( y  =  w  ->  (
w  =  x  ->  x  =  y )
)
65adantrd 456 . . . . . 6  |-  ( y  =  w  ->  (
( w  =  x  /\  ph )  ->  x  =  y )
)
76alrimiv 1619 . . . . 5  |-  ( y  =  w  ->  A. x
( ( w  =  x  /\  ph )  ->  x  =  y ) )
87a1d 24 . . . 4  |-  ( y  =  w  ->  (
w  e.  z  ->  A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) ) )
98spimev 1946 . . 3  |-  ( w  e.  z  ->  E. y A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) )
102, 9mpg 1537 . 2  |-  E. y A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
11 an12 774 . . . . . . 7  |-  ( ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
1211exbii 1571 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )
13 nfv 1607 . . . . . . 7  |-  F/ w
( x  e.  z  /\  ph )
14 elequ1 1690 . . . . . . . 8  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
1514anbi1d 687 . . . . . . 7  |-  ( w  =  x  ->  (
( w  e.  z  /\  ph )  <->  ( x  e.  z  /\  ph )
) )
1613, 15equsex 1907 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1712, 16bitr3i 244 . . . . 5  |-  ( E. w ( w  e.  z  /\  ( w  =  x  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1817bibi2i 306 . . . 4  |-  ( ( x  e.  y  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )  <->  ( x  e.  y  <->  ( x  e.  z  /\  ph )
) )
1918albii 1555 . . 3  |-  ( A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )  <->  A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) ) )
2019exbii 1571 . 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 201 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1529   E.wex 1530    = wceq 1625    e. wcel 1687
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1531  df-nf 1534  df-cleq 2279  df-clel 2282
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