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Mirrors > Home > ILE Home > Th. List > pweqb | GIF version |
Description: Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
Ref | Expression |
---|---|
pweqb | ⊢ (A = B ↔ 𝒫 A = 𝒫 B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwb 3943 | . . 3 ⊢ (A ⊆ B ↔ 𝒫 A ⊆ 𝒫 B) | |
2 | sspwb 3943 | . . 3 ⊢ (B ⊆ A ↔ 𝒫 B ⊆ 𝒫 A) | |
3 | 1, 2 | anbi12i 433 | . 2 ⊢ ((A ⊆ B ∧ B ⊆ A) ↔ (𝒫 A ⊆ 𝒫 B ∧ 𝒫 B ⊆ 𝒫 A)) |
4 | eqss 2954 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
5 | eqss 2954 | . 2 ⊢ (𝒫 A = 𝒫 B ↔ (𝒫 A ⊆ 𝒫 B ∧ 𝒫 B ⊆ 𝒫 A)) | |
6 | 3, 4, 5 | 3bitr4i 201 | 1 ⊢ (A = B ↔ 𝒫 A = 𝒫 B) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1242 ⊆ 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-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 df-sn 3373 |
This theorem is referenced by: (None) |
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