Theorem disjxsn

Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Assertion	disjxsn	|- Disj_ x e. { A } B

Proof

Step	Hyp	Ref	Expression
1		dfdisj2	|- ( Disj_ x e. { A } B <-> A. y E* x ( x e. { A } /\ y e. B ) )
2		moeq	|- E* x x = A
3		elsni	|- ( x e. { A } -> x = A )
4	3	adantr	|- ( ( x e. { A } /\ y e. B ) -> x = A )
5	4	moimi	|- ( E* x x = A -> E* x ( x e. { A } /\ y e. B ) )
6	2 5	ax-mp	|- E* x ( x e. { A } /\ y e. B )
7	1 6	mpgbir	|- Disj_ x e. { A } B