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Axiom ax-sep 3922
Description: The Axiom of Separation of IZF set theory. Axiom 6 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed, and with a 𝑦𝜑 condition replaced by a distinct variable constraint between 𝑦 and 𝜑).

The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with 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 2825. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

(Contributed by NM, 11-Sep-2006.)

Assertion
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 1435 . . . 4 wff 𝑥𝑦
4 vz . . . . . 6 setvar 𝑧
51, 4wel 1435 . . . . 5 wff 𝑥𝑧
6 wph . . . . 5 wff 𝜑
75, 6wa 102 . . . 4 wff (𝑥𝑧𝜑)
83, 7wb 103 . . 3 wff (𝑥𝑦 ↔ (𝑥𝑧𝜑))
98, 1wal 1283 . 2 wff 𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
109, 2wex 1422 1 wff 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff set class
This axiom is referenced by:  axsep2  3923  zfauscl  3924  bm1.3ii  3925  a9evsep  3926  axnul  3929  nalset  3934
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