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Axiom ax-pow 3900
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 3851).

The variant axpow2 3902 uses explicit subset notation. A version using class notation is pwex 3905. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-pow yz(w(w zw x) → z y)
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 1376 . . . . . 6 wff w z
4 vx . . . . . . 7 setvar x
51, 4wel 1376 . . . . . 6 wff w x
63, 5wi 4 . . . . 5 wff (w zw x)
76, 1wal 1226 . . . 4 wff w(w zw x)
8 vy . . . . 5 setvar y
92, 8wel 1376 . . . 4 wff z y
107, 9wi 4 . . 3 wff (w(w zw x) → z y)
1110, 2wal 1226 . 2 wff z(w(w zw x) → z y)
1211, 8wex 1363 1 wff yz(w(w zw x) → z y)
Colors of variables: wff set class
This axiom is referenced by:  zfpow  3901  axpow2  3902
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