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Axiom ax-pow 4668
Description: Axiom of Power Sets. An axiom of 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. The variant axpow2 4670 uses explicit subset notation. A version using class notation is pwex 4673. (Contributed by NM, 21-Jun-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 1939 . . . . . 6 wff 𝑤𝑧
4 vx . . . . . . 7 setvar 𝑥
51, 4wel 1939 . . . . . 6 wff 𝑤𝑥
63, 5wi 4 . . . . 5 wff (𝑤𝑧𝑤𝑥)
76, 1wal 1472 . . . 4 wff 𝑤(𝑤𝑧𝑤𝑥)
8 vy . . . . 5 setvar 𝑦
92, 8wel 1939 . . . 4 wff 𝑧𝑦
107, 9wi 4 . . 3 wff (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
1110, 2wal 1472 . 2 wff 𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
1211, 8wex 1694 1 wff 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
This axiom is referenced by:  zfpow  4669  axpow2  4670  bj-zfpow  31825
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