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Mirrors > Home > MPE Home > Th. List > pwexr | Structured version Visualization version GIF version |
Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5371. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
pwexr | ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unipw 5461 | . 2 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | uniexg 7759 | . 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V) | |
3 | 1, 2 | eqeltrrid 2844 | 1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 𝒫 cpw 4605 ∪ cuni 4912 |
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-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: pwexb 7785 pwuninel 8299 pwwf 9845 r1pw 9883 isfin3 10334 dis2ndc 23484 numufl 23939 bj-discrmoore 37094 |
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