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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 3354 | . . 3 ⊢ (x = A → 𝒫 x = 𝒫 A) | |
2 | 1 | eleq1d 2103 | . 2 ⊢ (x = A → (𝒫 x ∈ V ↔ 𝒫 A ∈ V)) |
3 | vex 2554 | . . 3 ⊢ x ∈ V | |
4 | 3 | pwex 3923 | . 2 ⊢ 𝒫 x ∈ V |
5 | 2, 4 | vtoclg 2607 | 1 ⊢ (A ∈ 𝑉 → 𝒫 A ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 Vcvv 2551 𝒫 cpw 3351 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-pw 3353 |
This theorem is referenced by: abssexg 3925 snexgOLD 3926 snexg 3927 pwel 3945 uniexb 4171 xpexg 4395 fabexg 5020 fopwdom 6246 |
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