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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 5231. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
pwexr | ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unipw 5308 | . 2 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | uniexg 7446 | . 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V) | |
3 | 1, 2 | eqeltrrid 2895 | 1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3441 𝒫 cpw 4497 ∪ cuni 4800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-uni 4801 |
This theorem is referenced by: pwexb 7468 pwuninel 7924 pwfi 8803 pwwf 9220 r1pw 9258 isfin3 9707 dis2ndc 22065 numufl 22520 bj-discrmoore 34526 |
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