Theorem reldisj

Description: Two ways of saying that two classes are disjoint, using the complement of B relative to a universe C . (Contributed by NM, 15-Feb-2007) (Proof shortened by Andrew Salmon, 26-Jun-2011) Avoid ax-12 . (Revised by Gino Giotto, 28-Jun-2024)

	Ref	Expression
Assertion	reldisj	|- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfss2	|- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
2		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
3		eleq1w	|- ( x = y -> ( x e. C <-> y e. C ) )
4	2 3	imbi12d	|- ( x = y -> ( ( x e. A -> x e. C ) <-> ( y e. A -> y e. C ) ) )
5	4	spw	|- ( A. x ( x e. A -> x e. C ) -> ( x e. A -> x e. C ) )
6		pm5.44	|- ( ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> ( x e. C /\ -. x e. B ) ) ) )
7		eldif	|- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
8	7	imbi2i	|- ( ( x e. A -> x e. ( C \ B ) ) <-> ( x e. A -> ( x e. C /\ -. x e. B ) ) )
9	6 8	bitr4di	|- ( ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
10	5 9	syl	|- ( A. x ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
11	1 10	sylbi	|- ( A C_ C -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
12	11	albidv	|- ( A C_ C -> ( A. x ( x e. A -> -. x e. B ) <-> A. x ( x e. A -> x e. ( C \ B ) ) ) )
13		disj1	|- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
14		dfss2	|- ( A C_ ( C \ B ) <-> A. x ( x e. A -> x e. ( C \ B ) ) )
15	12 13 14	3bitr4g	|- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) )