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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 5460 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
2 | sspwb 5460 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴) | |
3 | 1, 2 | anbi12i 628 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴)) |
4 | eqss 4011 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | eqss 4011 | . 2 ⊢ (𝒫 𝐴 = 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴)) | |
6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ⊆ wss 3963 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 |
This theorem is referenced by: psspwb 42246 |
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