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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. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) |
Ref | Expression |
---|---|
pwexg | ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 3362 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | eleq1d 2106 | . 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V)) |
3 | vex 2560 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | pwex 3932 | . 2 ⊢ 𝒫 𝑥 ∈ V |
5 | 2, 4 | vtoclg 2613 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 Vcvv 2557 𝒫 cpw 3359 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-pw 3361 |
This theorem is referenced by: abssexg 3934 snexgOLD 3935 snexg 3936 pwel 3954 uniexb 4205 xpexg 4452 fabexg 5077 fopwdom 6310 |
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