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Theorem axsep 3625
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 3616. 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 2608. 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 3628, 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 3668 shows (contradicting zfauscl 3628). However, as axsep2 3627 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 3626 from ax-rep 3616.

This theorem should not be referenced by any proof. Instead, use ax-sep 3626 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 1542 . . . 4
21axrep5 3621 . . 3
3 a9e 1656 . . . 4
4 equtr 1664 . . . . . . . 8
5 equcomi 1661 . . . . . . . 8
64, 5syl6 29 . . . . . . 7
76adantrd 447 . . . . . 6
87alrimiv 1820 . . . . 5
98eximi 1474 . . . 4
103, 9mp1i 11 . . 3
112, 10mpg 1450 . 2
12 an12 734 . . . . . . 7
1312exbii 1481 . . . . . 6
14 ax-17 1542 . . . . . . 7
15 elequ1 1669 . . . . . . . 8
1615anbi1d 679 . . . . . . 7
1714, 16equsex 1681 . . . . . 6
1813, 17bitr3i 240 . . . . 5
1918bibi2i 302 . . . 4
2019albii 1462 . . 3
2120exbii 1481 . 2
2211, 21mpbi 197 1
Colors of variables: wff set class
Syntax hints:   wi 4   wb 174   wa 357  wal 1442  wex 1447   wceq 1526   wcel 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1443  ax-6 1444  ax-7 1445  ax-gen 1446  ax-8 1530  ax-13 1534  ax-14 1535  ax-17 1542  ax-9 1557  ax-4 1563  ax-rep 3616
This theorem depends on definitions:  df-bi 175  df-an 359  df-ex 1448
Copyright terms: Public domain