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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 3578 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | eleq1d 2246 | . 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V)) |
3 | vpwex 4178 | . 2 ⊢ 𝒫 𝑥 ∈ V | |
4 | 2, 3 | vtoclg 2797 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 𝒫 cpw 3575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 |
This theorem is referenced by: pwexd 4180 abssexg 4181 pwex 4182 snexg 4183 pwel 4217 uniexb 4472 xpexg 4739 fabexg 5402 mapex 6651 pmvalg 6656 fopwdom 6833 ssenen 6848 restid2 12685 toponsspwpwg 13391 tgdom 13443 distop 13456 epttop 13461 cldval 13470 ntrfval 13471 clsfval 13472 neifval 13511 neif 13512 neival 13514 |
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