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Theorem axsep 4354
 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 4345. 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 ) so that it asserts the existence of a collection only if it is smaller than some other collection that already exists. This prevents Russell's paradox ru 3166. In some texts, this scheme is called "Aussonderung" or the Subset Axiom. The variable can appear free in the wff , which in textbooks is often written . To specify this in the Metamath language, we omit the distinct variable requirement (\$d) that not appear in . For a version using a class variable, see zfauscl 4357, which requires the Axiom of Extensionality as well as Separation for its derivation. If we omit the requirement that not occur in , we can derive a contradiction, as notzfaus 4403 shows (contradicting zfauscl 4357). However, as axsep2 4356 shows, we can eliminate the restriction that not occur in . Note: the distinct variable restriction that not occur in is actually redundant in this particular proof, but we keep it since its purpose is to demonstrate the derivation of the exact ax-sep 4355 from ax-rep 4345. This theorem should not be referenced by any proof. Instead, use ax-sep 4355 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
Distinct variable groups:   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem axsep
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1630 . . . 4
21axrep5 4350 . . 3
3 equtr 1696 . . . . . . . 8
4 equcomi 1693 . . . . . . . 8
53, 4syl6 32 . . . . . . 7
65adantrd 456 . . . . . 6
76alrimiv 1642 . . . . 5
87a1d 24 . . . 4
98spimev 1967 . . 3
102, 9mpg 1558 . 2
11 an12 774 . . . . . . 7
1211exbii 1593 . . . . . 6
13 nfv 1630 . . . . . . 7
14 elequ1 1730 . . . . . . . 8
1514anbi1d 687 . . . . . . 7
1613, 15equsex 2005 . . . . . 6
1712, 16bitr3i 244 . . . . 5
1817bibi2i 306 . . . 4
1918albii 1576 . . 3
2019exbii 1593 . 2
2110, 20mpbi 201 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wal 1550  wex 1551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-rep 4345 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555
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