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Axiom ax-pow 3977
Description: Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the power set of a given set 𝑥 i.e. contains every subset of 𝑥. This is Axiom 8 of [Crosilla] p. "Axioms of CZF and IZF" except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3928).

The variant axpow2 3979 uses explicit subset notation. A version using class notation is pwex 3985. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-pow 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Detailed syntax breakdown of Axiom ax-pow
StepHypRef Expression
1 vw . . . . . . 7 setvar 𝑤
2 vz . . . . . . 7 setvar 𝑧
31, 2wel 1437 . . . . . 6 wff 𝑤𝑧
4 vx . . . . . . 7 setvar 𝑥
51, 4wel 1437 . . . . . 6 wff 𝑤𝑥
63, 5wi 4 . . . . 5 wff (𝑤𝑧𝑤𝑥)
76, 1wal 1285 . . . 4 wff 𝑤(𝑤𝑧𝑤𝑥)
8 vy . . . . 5 setvar 𝑦
92, 8wel 1437 . . . 4 wff 𝑧𝑦
107, 9wi 4 . . 3 wff (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
1110, 2wal 1285 . 2 wff 𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
1211, 8wex 1424 1 wff 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff set class
This axiom is referenced by:  zfpow  3978  axpow2  3979
  Copyright terms: Public domain W3C validator