Theorem relssi

Description: Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998)

	Ref	Expression
Hypotheses	relssi.1	|- Rel A
	relssi.2	|- ( <. x , y >. e. A -> <. x , y >. e. B )
Assertion	relssi	|- A C_ B

Proof

Step	Hyp	Ref	Expression
1		relssi.1	|- Rel A
2		relssi.2	|- ( <. x , y >. e. A -> <. x , y >. e. B )
3		ssrel	|- ( Rel A -> ( A C_ B <-> A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) ) )
4	1 3	ax-mp	|- ( A C_ B <-> A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) )
5	2	ax-gen	|- A. y ( <. x , y >. e. A -> <. x , y >. e. B )
6	4 5	mpgbir	|- A C_ B