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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 | uniexb 6866 | . 2 ⊢ (𝒫 𝐴 ∈ V ↔ ∪ 𝒫 𝐴 ∈ V) | |
2 | unipw 4845 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 2 | eleq1i 2679 | . 2 ⊢ (∪ 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V) |
4 | 1, 3 | bitr2i 264 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 ∪ cuni 4372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-pr 4128 df-uni 4373 |
This theorem is referenced by: pwuninel 7288 2pwuninel 8000 pwfi 8144 pwwf 8553 ranklim 8590 r1pw 8591 r1pwALT 8592 isfin3 9001 isf34lem6 9085 isfin1-2 9090 pwfseqlem4 9363 pwfseqlem5 9364 gchpwdom 9371 hargch 9374 dis2ndc 21073 numufl 21529 |
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