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Theorem axsep 4545
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 4536. 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 3298. 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 4548, 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 4599 shows (contradicting zfauscl 4548). However, as axsep2 4547 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 4546 from ax-rep 4536.

This theorem should not be referenced by any proof. Instead, use ax-sep 4546 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 1755 . . . 4  |-  F/ y ( w  =  x  /\  ph )
21axrep5 4541 . . 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 1850 . . . . . . . 8  |-  ( y  =  w  ->  (
w  =  x  -> 
y  =  x ) )
4 equcomi 1847 . . . . . . . 8  |-  ( y  =  x  ->  x  =  y )
53, 4syl6 34 . . . . . . 7  |-  ( y  =  w  ->  (
w  =  x  ->  x  =  y )
)
65adantrd 469 . . . . . 6  |-  ( y  =  w  ->  (
( w  =  x  /\  ph )  ->  x  =  y )
)
76alrimiv 1767 . . . . 5  |-  ( y  =  w  ->  A. x
( ( w  =  x  /\  ph )  ->  x  =  y ) )
87a1d 26 . . . 4  |-  ( y  =  w  ->  (
w  e.  z  ->  A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) ) )
98spimev 2068 . . 3  |-  ( w  e.  z  ->  E. y A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) )
102, 9mpg 1665 . 2  |-  E. y A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
11 an12 804 . . . . . . 7  |-  ( ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
1211exbii 1712 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )
13 nfv 1755 . . . . . . 7  |-  F/ w
( x  e.  z  /\  ph )
14 elequ1 1875 . . . . . . . 8  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
1514anbi1d 709 . . . . . . 7  |-  ( w  =  x  ->  (
( w  e.  z  /\  ph )  <->  ( x  e.  z  /\  ph )
) )
1613, 15equsex 2095 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1712, 16bitr3i 254 . . . . 5  |-  ( E. w ( w  e.  z  /\  ( w  =  x  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1817bibi2i 314 . . . 4  |-  ( ( x  e.  y  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )  <->  ( x  e.  y  <->  ( x  e.  z  /\  ph )
) )
1918albii 1685 . . 3  |-  ( A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )  <->  A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) ) )
2019exbii 1712 . 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 211 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-rep 4536
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662
This theorem is referenced by: (None)
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