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Axiom ax-sep 5094
Description: Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories.

It was derived as axsep 5093 above and is therefore redundant in ZF set theory, which contains ax-rep 5081 as an axiom (contrary to Zermelo set theory). We state it as a separate axiom here so that some of its uses can be identified more easily. Some textbooks present the axiom scheme of separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger axiom scheme of replacement (which is not part of Zermelo set theory).

The axiom scheme of separation is a weak form of Frege's axiom scheme of (unrestricted) comprehension, in that it conditions it with the condition 𝑥𝑧, 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 3705. In some texts, this scheme is called "Aussonderung" (German for "separation") or "Subset Axiom".

The variable 𝑥 can occur in the formula 𝜑, which in textbooks is often written 𝜑(𝑥). To specify this in the Metamath language, we omit the distinct variable condition ($d) that 𝑥 not occur in 𝜑.

For a version using a class variable, see zfauscl 5096, which requires the axiom of extensionality as well as the axiom scheme of separation for its derivation.

If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 5153 shows (showing the necessity of that condition in zfauscl 5096).

Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)

Ref Expression
ax-sep 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Axiom ax-sep
StepHypRef Expression
1 vx . . . . 5 setvar 𝑥
2 vy . . . . 5 setvar 𝑦
31, 2wel 2082 . . . 4 wff 𝑥𝑦
4 vz . . . . . 6 setvar 𝑧
51, 4wel 2082 . . . . 5 wff 𝑥𝑧
6 wph . . . . 5 wff 𝜑
75, 6wa 396 . . . 4 wff (𝑥𝑧𝜑)
83, 7wb 207 . . 3 wff (𝑥𝑦 ↔ (𝑥𝑧𝜑))
98, 1wal 1520 . 2 wff 𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
109, 2wex 1761 1 wff 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
This axiom is referenced by:  axsepg  5095  zfauscl  5096  bm1.3ii  5097  ax6vsep  5098  axnul  5100  nalset  5108  bj-zfauscl  33814  bj-bm1.3ii  33955
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