Theorem ltnegcon1d

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 )
	ltnegcon1d.3	|- ( ph -> -u A < B )
Assertion	ltnegcon1d	|- ( ph -> -u B < A )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		ltnegcon1d.3	|- ( ph -> -u A < B )
4		ltnegcon1	|- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )
5	1 2 4	syl2anc	|- ( ph -> ( -u A < B <-> -u B < A ) )
6	3 5	mpbid	|- ( ph -> -u B < A )