Theorem compleq 3752

Description: Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.)

Assertion

Ref	Expression
compleq	|- ( A = B <-> ( _V \ A ) = ( _V \ B ) )

Proof of Theorem compleq

Step	Hyp	Ref	Expression
1		complss 3751	. . 3 |- ( A C_ B <-> ( _V \ B ) C_ ( _V \ A ) )
2		complss 3751	. . 3 |- ( B C_ A <-> ( _V \ A ) C_ ( _V \ B ) )
3	1, 2	anbi12ci 734	. 2 |- ( ( A C_ B /\ B C_ A ) <-> ( ( _V \ A ) C_ ( _V \ B ) /\ ( _V \ B ) C_ ( _V \ A ) ) )
4		eqss 3618	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3618	. 2 |- ( ( _V \ A ) = ( _V \ B ) <-> ( ( _V \ A ) C_ ( _V \ B ) /\ ( _V \ B ) C_ ( _V \ A ) ) )
6	3, 4, 5	3bitr4i 292	1 |- ( A = B <-> ( _V \ A ) = ( _V \ B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   _Vcvv 3200   \ cdif 3571   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by: (None)