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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 5279 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
2 | pwexr 7487 | . 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V) | |
3 | 1, 2 | impbii 211 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 Vcvv 3494 𝒫 cpw 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-pw 4541 df-sn 4568 df-pr 4570 df-uni 4839 |
This theorem is referenced by: 2pwuninel 8672 ranklim 9273 r1pwALT 9275 isf34lem6 9802 isfin1-2 9807 pwfseqlem4 10084 pwfseqlem5 10085 gchpwdom 10092 hargch 10095 numufl 22523 |
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