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Mirrors > Home > MPE Home > Th. List > pwexb | Structured version Visualization version GIF version |
Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
pwexb | ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 5244 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
2 | pwexr 7467 | . 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V) | |
3 | 1, 2 | impbii 212 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 Vcvv 3441 𝒫 cpw 4497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-uni 4801 |
This theorem is referenced by: 2pwuninel 8656 ranklim 9257 r1pwALT 9259 isf34lem6 9791 isfin1-2 9796 pwfseqlem4 10073 pwfseqlem5 10074 gchpwdom 10081 hargch 10084 numufl 22520 |
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