Theorem complss 3751

Description: Complementation reverses inclusion. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 19-Mar-2021.)

Assertion

Ref	Expression
complss	|- ( A C_ B <-> ( _V \ B ) C_ ( _V \ A ) )

Proof of Theorem complss

Step	Hyp	Ref	Expression
1		sscon 3744	. 2 |- ( A C_ B -> ( _V \ B ) C_ ( _V \ A ) )
2		sscon 3744	. . 3 |- ( ( _V \ B ) C_ ( _V \ A ) -> ( _V \ ( _V \ A ) ) C_ ( _V \ ( _V \ B ) ) )
3		ddif 3742	. . 3 |- ( _V \ ( _V \ A ) ) = A
4		ddif 3742	. . 3 |- ( _V \ ( _V \ B ) ) = B
5	2, 3, 4	3sstr3g 3645	. 2 |- ( ( _V \ B ) C_ ( _V \ A ) -> A C_ B )
6	1, 5	impbii 199	1 |- ( A C_ B <-> ( _V \ B ) C_ ( _V \ A ) )

Syntax hints:   <-> wb 196   _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:  compleq  3752