Theorem bj-sscon 33014

Description: Contraposition law for relative subsets. Relative and generalized version of ssconb 3743, which it can shorten, as well as conss2 38647. (Contributed by BJ, 11-Nov-2021.)

Assertion

Ref	Expression
bj-sscon	|- ( ( A i^i V ) C_ ( V \ B ) <-> ( B i^i V ) C_ ( V \ A ) )

Proof of Theorem bj-sscon

Step	Hyp	Ref	Expression
1		incom 3805	. . . 4 |- ( A i^i B ) = ( B i^i A )
2	1	ineq1i 3810	. . 3 |- ( ( A i^i B ) i^i V ) = ( ( B i^i A ) i^i V )
3	2	eqeq1i 2627	. 2 |- ( ( ( A i^i B ) i^i V ) = (/) <-> ( ( B i^i A ) i^i V ) = (/) )
4		bj-disj2r 33013	. 2 |- ( ( A i^i V ) C_ ( V \ B ) <-> ( ( A i^i B ) i^i V ) = (/) )
5		bj-disj2r 33013	. 2 |- ( ( B i^i V ) C_ ( V \ A ) <-> ( ( B i^i A ) i^i V ) = (/) )
6	3, 4, 5	3bitr4i 292	1 |- ( ( A i^i V ) C_ ( V \ B ) <-> ( B i^i V ) C_ ( V \ A ) )

Syntax hints:   <-> wb 196   = wceq 1483   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915

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-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by: (None)