Theorem lenegcon2d

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

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	lenegcon2d.3	|- ( ph -> A <_ -u B )
Assertion	lenegcon2d	|- ( ph -> B <_ -u A )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		lenegcon2d.3	|- ( ph -> A <_ -u B )
4		lenegcon2	|- ( ( 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 )