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Theorem axsep 3490
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 3481. 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 2495. 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 3493, which requires the Axiom of Extensionality as well as Replacement for its derivation.

If we omit the requirement that not occur in , we can derive a contradiction, as notzfaus 3533 shows (contradicting zfauscl 3493). However, as axsep2 3492 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 3491 from ax-rep 3481.

This theorem should not be referenced by any proof. Instead, use ax-sep 3491 below so that the uses of the Axiom of Separation can be more easily identified.

Assertion
Ref Expression
axsep
Distinct variable groups:   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem axsep
StepHypRef Expression
1 ax-17 1430 . . . 4
21axrep5 3486 . . 3
3 a9e 1544 . . . 4
4 equtr 1552 . . . . . . . 8
5 equcomi 1549 . . . . . . . 8
64, 5syl6 29 . . . . . . 7
76adantrd 447 . . . . . 6
87alrimiv 1708 . . . . 5
98eximi 1362 . . . 4
103, 9mp1i 11 . . 3
112, 10mpg 1338 . 2
12 an12 734 . . . . . . 7
1312exbii 1369 . . . . . 6
14 ax-17 1430 . . . . . . 7
15 elequ1 1557 . . . . . . . 8
1615anbi1d 679 . . . . . . 7
1714, 16equsex 1569 . . . . . 6
1813, 17bitr3i 240 . . . . 5
1918bibi2i 302 . . . 4
2019albii 1350 . . 3
2120exbii 1369 . 2
2211, 21mpbi 197 1
Colors of variables: wff set class
Syntax hints:   wi 4   wb 174   wa 357  wal 1330  wex 1335   wceq 1414   wcel 1416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1331  ax-6 1332  ax-7 1333  ax-gen 1334  ax-8 1418  ax-13 1422  ax-14 1423  ax-17 1430  ax-9 1445  ax-4 1451  ax-rep 3481
This theorem depends on definitions:  df-bi 175  df-an 359  df-ex 1336
Copyright terms: Public domain