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Theorem pweqb 3933
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 3926 . . 3 (AB ↔ 𝒫 A ⊆ 𝒫 B)
2 sspwb 3926 . . 3 (BA ↔ 𝒫 B ⊆ 𝒫 A)
31, 2anbi12i 436 . 2 ((AB BA) ↔ (𝒫 A ⊆ 𝒫 B 𝒫 B ⊆ 𝒫 A))
4 eqss 2937 . 2 (A = B ↔ (AB BA))
5 eqss 2937 . 2 (𝒫 A = 𝒫 B ↔ (𝒫 A ⊆ 𝒫 B 𝒫 B ⊆ 𝒫 A))
63, 4, 53bitr4i 201 1 (A = B ↔ 𝒫 A = 𝒫 B)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1228  wss 2894  𝒫 cpw 3334
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356
This theorem is referenced by: (None)
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