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Mirrors > Home > ILE Home > Th. List > pwexg | GIF version |
Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.) |
Ref | Expression |
---|---|
pwexg | ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 3569 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | eleq1d 2239 | . 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V)) |
3 | vpwex 4165 | . 2 ⊢ 𝒫 𝑥 ∈ V | |
4 | 2, 3 | vtoclg 2790 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 Vcvv 2730 𝒫 cpw 3566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-pw 3568 |
This theorem is referenced by: pwexd 4167 abssexg 4168 pwex 4169 snexg 4170 pwel 4203 uniexb 4458 xpexg 4725 fabexg 5385 mapex 6632 pmvalg 6637 fopwdom 6814 ssenen 6829 restid2 12588 toponsspwpwg 12814 tgdom 12866 distop 12879 epttop 12884 cldval 12893 ntrfval 12894 clsfval 12895 neifval 12934 neif 12935 neival 12937 |
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