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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 3546 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | eleq1d 2226 | . 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V)) |
3 | vpwex 4140 | . 2 ⊢ 𝒫 𝑥 ∈ V | |
4 | 2, 3 | vtoclg 2772 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 Vcvv 2712 𝒫 cpw 3543 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-in 3108 df-ss 3115 df-pw 3545 |
This theorem is referenced by: pwexd 4142 abssexg 4143 pwex 4144 snexg 4145 pwel 4178 uniexb 4433 xpexg 4700 fabexg 5357 mapex 6599 pmvalg 6604 fopwdom 6781 ssenen 6796 restid2 12371 toponsspwpwg 12431 tgdom 12483 distop 12496 epttop 12501 cldval 12510 ntrfval 12511 clsfval 12512 neifval 12551 neif 12552 neival 12554 |
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