Theorem pweqb 5427

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 5420	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2		sspwb 5420	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
3	1, 2	anbi12i 639	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
4		eqss 3954	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		eqss 3954	. 2 ⊢ (𝒫 𝐴 = 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
6	3, 4, 5	3bitr4i 306	1 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563   ⊆ wss 3907  𝒫 cpw 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-ss 3924  df-pw 4560  df-sn 4586  df-pr 4588

This theorem is referenced by:  psspwb  42854