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Theorem pweqb 5074
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.)
Assertion
Ref Expression
pweqb (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)

Proof of Theorem pweqb
StepHypRef Expression
1 sspwb 5066 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 sspwb 5066 . . 3 (𝐵𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
31, 2anbi12i 735 . 2 ((𝐴𝐵𝐵𝐴) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
4 eqss 3759 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3759 . 2 (𝒫 𝐴 = 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
63, 4, 53bitr4i 292 1 (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wss 3715  𝒫 cpw 4302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324
This theorem is referenced by: (None)
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