Theorem pweqb 3950

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 3943	. . 3 ⊢ (A ⊆ B ↔ 𝒫 A ⊆ 𝒫 B)
2		sspwb 3943	. . 3 ⊢ (B ⊆ A ↔ 𝒫 B ⊆ 𝒫 A)
3	1, 2	anbi12i 433	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) ↔ (𝒫 A ⊆ 𝒫 B ∧ 𝒫 B ⊆ 𝒫 A))
4		eqss 2954	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 2954	. 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 1242   ⊆ wss 2911  𝒫 cpw 3351

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by: (None)