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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 3513 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | eleq1d 2208 | . 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V)) |
3 | vpwex 4103 | . 2 ⊢ 𝒫 𝑥 ∈ V | |
4 | 2, 3 | vtoclg 2746 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 𝒫 cpw 3510 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-in 3077 df-ss 3084 df-pw 3512 |
This theorem is referenced by: pwexd 4105 abssexg 4106 pwex 4107 snexg 4108 pwel 4140 uniexb 4394 xpexg 4653 fabexg 5310 mapex 6548 pmvalg 6553 fopwdom 6730 ssenen 6745 restid2 12129 toponsspwpwg 12189 tgdom 12241 distop 12254 epttop 12259 cldval 12268 ntrfval 12269 clsfval 12270 neifval 12309 neif 12310 neival 12312 |
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