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Mirrors > Home > MPE Home > Th. List > pweqb | Structured version Visualization version 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 4844 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
2 | sspwb 4844 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴) | |
3 | 1, 2 | anbi12i 729 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴)) |
4 | eqss 3583 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | eqss 3583 | . 2 ⊢ (𝒫 𝐴 = 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴)) | |
6 | 3, 4, 5 | 3bitr4i 291 | 1 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ⊆ wss 3540 𝒫 cpw 4108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-pr 4128 |
This theorem is referenced by: (None) |
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