Theorem disjel

Description: A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015)

	Ref	Expression
Assertion	disjel	|- ( ( ( A i^i B ) = (/) /\ C e. A ) -> -. C e. B )

Proof

Step	Hyp	Ref	Expression
1		disj3	|- ( ( A i^i B ) = (/) <-> A = ( A \ B ) )
2		eleq2	|- ( A = ( A \ B ) -> ( C e. A <-> C e. ( A \ B ) ) )
3		eldifn	|- ( C e. ( A \ B ) -> -. C e. B )
4	2 3	syl6bi	|- ( A = ( A \ B ) -> ( C e. A -> -. C e. B ) )
5	1 4	sylbi	|- ( ( A i^i B ) = (/) -> ( C e. A -> -. C e. B ) )
6	5	imp	|- ( ( ( A i^i B ) = (/) /\ C e. A ) -> -. C e. B )