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Mirrors > Home > ILE Home > Th. List > elpw2g | GIF version |
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.) |
Ref | Expression |
---|---|
elpw2g | ⊢ (B ∈ 𝑉 → (A ∈ 𝒫 B ↔ A ⊆ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 3360 | . 2 ⊢ (A ∈ 𝒫 B → A ⊆ B) | |
2 | ssexg 3887 | . . . 4 ⊢ ((A ⊆ B ∧ B ∈ 𝑉) → A ∈ V) | |
3 | elpwg 3359 | . . . . 5 ⊢ (A ∈ V → (A ∈ 𝒫 B ↔ A ⊆ B)) | |
4 | 3 | biimparc 283 | . . . 4 ⊢ ((A ⊆ B ∧ A ∈ V) → A ∈ 𝒫 B) |
5 | 2, 4 | syldan 266 | . . 3 ⊢ ((A ⊆ B ∧ B ∈ 𝑉) → A ∈ 𝒫 B) |
6 | 5 | expcom 109 | . 2 ⊢ (B ∈ 𝑉 → (A ⊆ B → A ∈ 𝒫 B)) |
7 | 1, 6 | impbid2 131 | 1 ⊢ (B ∈ 𝑉 → (A ∈ 𝒫 B ↔ A ⊆ B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 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-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 |
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: elpw2 3902 pwnss 3903 |
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