Theorem ltnegcon2d

Description: Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	ltnegcon2d.3	|- ( ph -> A < -u B )
Assertion	ltnegcon2d	|- ( ph -> B < -u A )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		ltnegcon2d.3	|- ( ph -> A < -u B )
4		ltnegcon2	|- ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> B < -u A ) )
5	1 2 4	syl2anc	|- ( ph -> ( A < -u B <-> B < -u A ) )
6	3 5	mpbid	|- ( ph -> B < -u A )