Theorem posdifd

Description: Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
Assertion	posdifd	|- ( ph -> ( A < B <-> 0 < ( B - A ) ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		posdif	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( A < B <-> 0 < ( B - A ) ) )