Metamath Proof Explorer


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 )