Theorem ecin0

Description: Two ways of saying that the coset of A and the coset of B have no elements in common. (Contributed by Peter Mazsa, 1-Dec-2018)

	Ref	Expression
Assertion	ecin0	|- ( ( A e. V /\ B e. W ) -> ( ( [ A ] R i^i [ B ] R ) = (/) <-> A. x ( A R x -> -. B R x ) ) )

Proof

Step	Hyp	Ref	Expression
1		disj1	|- ( ( [ A ] R i^i [ B ] R ) = (/) <-> A. x ( x e. [ A ] R -> -. x e. [ B ] R ) )
2		elecg	|- ( ( x e. _V /\ A e. V ) -> ( x e. [ A ] R <-> A R x ) )
3	2	el2v1	|- ( A e. V -> ( x e. [ A ] R <-> A R x ) )
4	3	adantr	|- ( ( A e. V /\ B e. W ) -> ( x e. [ A ] R <-> A R x ) )
5		elecALTV	|- ( ( B e. W /\ x e. _V ) -> ( x e. [ B ] R <-> B R x ) )
6	5	elvd	|- ( B e. W -> ( x e. [ B ] R <-> B R x ) )
7	6	adantl	|- ( ( A e. V /\ B e. W ) -> ( x e. [ B ] R <-> B R x ) )
8	7	notbid	|- ( ( A e. V /\ B e. W ) -> ( -. x e. [ B ] R <-> -. B R x ) )
9	4 8	imbi12d	|- ( ( A e. V /\ B e. W ) -> ( ( x e. [ A ] R -> -. x e. [ B ] R ) <-> ( A R x -> -. B R x ) ) )
10	9	albidv	|- ( ( A e. V /\ B e. W ) -> ( A. x ( x e. [ A ] R -> -. x e. [ B ] R ) <-> A. x ( A R x -> -. B R x ) ) )
11	1 10	syl5bb	|- ( ( A e. V /\ B e. W ) -> ( ( [ A ] R i^i [ B ] R ) = (/) <-> A. x ( A R x -> -. B R x ) ) )