Metamath Proof Explorer


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 )