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Mirrors > Home > ILE Home > Th. List > ax-pow | Unicode version |
Description: Axiom of Power Sets. An
axiom of Intuitionistic Zermelo-Fraenkel set
theory. It states that a set exists that includes the power set
of a given set
i.e. contains every subset of . 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 4110).
The variant axpow2 4162 uses explicit subset notation. A version using class notation is pwex 4169. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ax-pow |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vw | . . . . . . 7 | |
2 | vz | . . . . . . 7 | |
3 | 1, 2 | wel 2142 | . . . . . 6 |
4 | vx | . . . . . . 7 | |
5 | 1, 4 | wel 2142 | . . . . . 6 |
6 | 3, 5 | wi 4 | . . . . 5 |
7 | 6, 1 | wal 1346 | . . . 4 |
8 | vy | . . . . 5 | |
9 | 2, 8 | wel 2142 | . . . 4 |
10 | 7, 9 | wi 4 | . . 3 |
11 | 10, 2 | wal 1346 | . 2 |
12 | 11, 8 | wex 1485 | 1 |
Colors of variables: wff set class |
This axiom is referenced by: zfpow 4161 axpow2 4162 |
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