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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 5365. (Contributed by NM, 11-Nov-2003.) |
| Ref | Expression |
|---|---|
| pwexr | ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unipw 5455 | . 2 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
| 2 | uniexg 7760 | . 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V) | |
| 3 | 1, 2 | eqeltrrid 2846 | 1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 𝒫 cpw 4600 ∪ cuni 4907 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 |
| This theorem is referenced by: pwexb 7786 pwuninel 8300 pwwf 9847 r1pw 9885 isfin3 10336 dis2ndc 23468 numufl 23923 bj-discrmoore 37112 |
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