Theorem pweqb 1867

Description: Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18.

Assertion

Ref	Expression
pweqb	⊢ (A = B ↔ ℘A = ℘B)

Proof of Theorem pweqb

Step	Hyp	Ref	Expression
1		sspwb 1863	. . 3 ⊢ (A ⊆ B ↔ ℘A ⊆ ℘B)
2		sspwb 1863	. . 3 ⊢ (B ⊆ A ↔ ℘B ⊆ ℘A)
3	1, 2	anbi12i 369	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) ↔ (℘A ⊆ ℘B ∧ ℘B ⊆ ℘A))
4		eqss 1516	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 1516	. 2 ⊢ (℘A = ℘B ↔ (℘A ⊆ ℘B ∧ ℘B ⊆ ℘A))
6	3, 4, 5	3bitr4 158	1 ⊢ (A = B ↔ ℘A = ℘B)

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ⊆ wss 1487  ℘cpw 1798

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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