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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 5377 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
| 2 | pwexr 7786 | . 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V) | |
| 3 | 1, 2 | impbii 209 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2107 Vcvv 3479 𝒫 cpw 4599 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-ss 3967 df-pw 4601 df-sn 4626 df-pr 4628 df-uni 4907 | 
| This theorem is referenced by: 2pwuninel 9173 ranklim 9885 r1pwALT 9887 isf34lem6 10421 isfin1-2 10426 pwfseqlem4 10703 pwfseqlem5 10704 gchpwdom 10711 hargch 10714 numufl 23924 | 
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