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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 5268. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
pwexr | ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unipw 5345 | . 2 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | uniexg 7468 | . 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V) | |
3 | 1, 2 | eqeltrrid 2920 | 1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3496 𝒫 cpw 4541 ∪ cuni 4840 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-pw 4543 df-sn 4570 df-pr 4572 df-uni 4841 |
This theorem is referenced by: pwexb 7490 pwuninel 7943 pwfi 8821 pwwf 9238 r1pw 9276 isfin3 9720 dis2ndc 22070 numufl 22525 bj-discrmoore 34405 |
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