Theorem sscon34b

Description: Relative complementation reverses inclusion of subclasses. Relativized version of complss . (Contributed by RP, 3-Jun-2021)

	Ref	Expression
Assertion	sscon34b	|- ( ( A C_ C /\ B C_ C ) -> ( A C_ B <-> ( C \ B ) C_ ( C \ A ) ) )

Proof

Step	Hyp	Ref	Expression
1		sscon	|- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )
2		sscon	|- ( ( C \ B ) C_ ( C \ A ) -> ( C \ ( C \ A ) ) C_ ( C \ ( C \ B ) ) )
3		dfss4	|- ( A C_ C <-> ( C \ ( C \ A ) ) = A )
4	3	biimpi	|- ( A C_ C -> ( C \ ( C \ A ) ) = A )
5	4	adantr	|- ( ( A C_ C /\ B C_ C ) -> ( C \ ( C \ A ) ) = A )
6		dfss4	|- ( B C_ C <-> ( C \ ( C \ B ) ) = B )
7	6	biimpi	|- ( B C_ C -> ( C \ ( C \ B ) ) = B )
8	7	adantl	|- ( ( A C_ C /\ B C_ C ) -> ( C \ ( C \ B ) ) = B )
9	5 8	sseq12d	|- ( ( A C_ C /\ B C_ C ) -> ( ( C \ ( C \ A ) ) C_ ( C \ ( C \ B ) ) <-> A C_ B ) )
10	2 9	syl5ib	|- ( ( A C_ C /\ B C_ C ) -> ( ( C \ B ) C_ ( C \ A ) -> A C_ B ) )
11	1 10	impbid2	|- ( ( A C_ C /\ B C_ C ) -> ( A C_ B <-> ( C \ B ) C_ ( C \ A ) ) )