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

Step	Hyp	Ref	Expression
1		sspwb 3926	. . 3 ⊢ (A ⊆ B ↔ 𝒫 A ⊆ 𝒫 B)
2		sspwb 3926	. . 3 ⊢ (B ⊆ A ↔ 𝒫 B ⊆ 𝒫 A)
3	1, 2	anbi12i 436	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) ↔ (𝒫 A ⊆ 𝒫 B ∧ 𝒫 B ⊆ 𝒫 A))
4		eqss 2937	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 2937	. 2 ⊢ (𝒫 A = 𝒫 B ↔ (𝒫 A ⊆ 𝒫 B ∧ 𝒫 B ⊆ 𝒫 A))
6	3, 4, 5	3bitr4i 201	1 ⊢ (A = B ↔ 𝒫 A = 𝒫 B)

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)