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Theorem axsep 3744
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 3735. 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 2687. 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 3747, 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 3786 shows (contradicting zfauscl 3747). However, as axsep2 3746 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 3745 from ax-rep 3735.

This theorem should not be referenced by any proof. Instead, use ax-sep 3745 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
StepHypRef Expression
1 ax-17 1540 . . . 4  |-  ( ( w  =  x  /\  ph )  ->  A. y
( w  =  x  /\  ph ) )
21axrep5 3740 . . 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 a9e 1691 . . . 4  |-  E. y 
y  =  w
4 equtr 1700 . . . . . . . 8  |-  ( y  =  w  ->  (
w  =  x  -> 
y  =  x ) )
5 equcomi 1696 . . . . . . . 8  |-  ( y  =  x  ->  x  =  y )
64, 5syl6 29 . . . . . . 7  |-  ( y  =  w  ->  (
w  =  x  ->  x  =  y )
)
76adantrd 449 . . . . . 6  |-  ( y  =  w  ->  (
( w  =  x  /\  ph )  ->  x  =  y )
)
87alrimiv 1869 . . . . 5  |-  ( y  =  w  ->  A. x
( ( w  =  x  /\  ph )  ->  x  =  y ) )
98eximi 1484 . . . 4  |-  ( E. y  y  =  w  ->  E. y A. x
( ( w  =  x  /\  ph )  ->  x  =  y ) )
103, 9mp1i 11 . . 3  |-  ( w  e.  z  ->  E. y A. x ( ( w  =  x  /\  ph )  ->  x  =  y ) )
112, 10mpg 1459 . 2  |-  E. y A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
12 an12 736 . . . . . . 7  |-  ( ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )
1312exbii 1490 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )
14 ax-17 1540 . . . . . . 7  |-  ( ( x  e.  z  /\  ph )  ->  A. w
( x  e.  z  /\  ph ) )
15 elequ1 1705 . . . . . . . 8  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
1615anbi1d 680 . . . . . . 7  |-  ( w  =  x  ->  (
( w  e.  z  /\  ph )  <->  ( x  e.  z  /\  ph )
) )
1714, 16equsex 1722 . . . . . 6  |-  ( E. w ( w  =  x  /\  ( w  e.  z  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1813, 17bitr3i 240 . . . . 5  |-  ( E. w ( w  e.  z  /\  ( w  =  x  /\  ph ) )  <->  ( x  e.  z  /\  ph )
)
1918bibi2i 302 . . . 4  |-  ( ( x  e.  y  <->  E. w
( w  e.  z  /\  ( w  =  x  /\  ph )
) )  <->  ( x  e.  y  <->  ( x  e.  z  /\  ph )
) )
2019albii 1471 . . 3  |-  ( A. x ( x  e.  y  <->  E. w ( w  e.  z  /\  (
w  =  x  /\  ph ) ) )  <->  A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) ) )
2120exbii 1490 . 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 )
) )
2211, 21mpbi 197 1  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    /\ wa 356   A.wal 1451   E.wex 1456    = wceq 1531    e. wcel 1533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1452  ax-6 1453  ax-7 1454  ax-gen 1455  ax-8 1535  ax-13 1537  ax-14 1538  ax-17 1540  ax-9 1594  ax-4 1601  ax-rep 3735
This theorem depends on definitions:  df-bi 175  df-an 358  df-ex 1457
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