Theorem pssdifcom2

Description: Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014)

	Ref	Expression
Assertion	pssdifcom2	|- ( ( A C_ C /\ B C_ C ) -> ( B C. ( C \ A ) <-> A C. ( C \ B ) ) )

Proof

Step	Hyp	Ref	Expression
1		ssconb	|- ( ( B C_ C /\ A C_ C ) -> ( B C_ ( C \ A ) <-> A C_ ( C \ B ) ) )
2	1	ancoms	|- ( ( A C_ C /\ B C_ C ) -> ( B C_ ( C \ A ) <-> A C_ ( C \ B ) ) )
3		difcom	|- ( ( C \ A ) C_ B <-> ( C \ B ) C_ A )
4	3	notbii	|- ( -. ( C \ A ) C_ B <-> -. ( C \ B ) C_ A )
5	4	a1i	|- ( ( A C_ C /\ B C_ C ) -> ( -. ( C \ A ) C_ B <-> -. ( C \ B ) C_ A ) )
6	2 5	anbi12d	|- ( ( A C_ C /\ B C_ C ) -> ( ( B C_ ( C \ A ) /\ -. ( C \ A ) C_ B ) <-> ( A C_ ( C \ B ) /\ -. ( C \ B ) C_ A ) ) )
7		dfpss3	|- ( B C. ( C \ A ) <-> ( B C_ ( C \ A ) /\ -. ( C \ A ) C_ B ) )
8		dfpss3	|- ( A C. ( C \ B ) <-> ( A C_ ( C \ B ) /\ -. ( C \ B ) C_ A ) )
9	6 7 8	3bitr4g	|- ( ( A C_ C /\ B C_ C ) -> ( B C. ( C \ A ) <-> A C. ( C \ B ) ) )