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Theorem pweqb 5467
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 5460 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 sspwb 5460 . . 3 (𝐵𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
31, 2anbi12i 628 . 2 ((𝐴𝐵𝐵𝐴) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
4 eqss 4011 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 4011 . 2 (𝒫 𝐴 = 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
63, 4, 53bitr4i 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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