Theorem disjss1f

Description: A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017)

	Ref	Expression
Hypotheses	disjss1f.1	|- F/_ x A
	disjss1f.2	|- F/_ x B
Assertion	disjss1f	|- ( A C_ B -> ( Disj_ x e. B C -> Disj_ x e. A C ) )

Proof

Step	Hyp	Ref	Expression
1		disjss1f.1	|- F/_ x A
2		disjss1f.2	|- F/_ x B
3	1 2	ssrmof	|- ( A C_ B -> ( E* x e. B y e. C -> E* x e. A y e. C ) )
4	3	alimdv	|- ( A C_ B -> ( A. y E* x e. B y e. C -> A. y E* x e. A y e. C ) )
5		df-disj	|- ( Disj_ x e. B C <-> A. y E* x e. B y e. C )
6		df-disj	|- ( Disj_ x e. A C <-> A. y E* x e. A y e. C )
7	4 5 6	3imtr4g	|- ( A C_ B -> ( Disj_ x e. B C -> Disj_ x e. A C ) )