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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 5384 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
2 | pwexr 7784 | . 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 2106 Vcvv 3478 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 |
This theorem is referenced by: 2pwuninel 9171 ranklim 9882 r1pwALT 9884 isf34lem6 10418 isfin1-2 10423 pwfseqlem4 10700 pwfseqlem5 10701 gchpwdom 10708 hargch 10711 numufl 23939 |
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