Metamath Proof Explorer

Theorem ltnegcon1i

Description: Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999)

Ref Expression
Hypotheses lt2.1 
|- A e. RR
lt2.2 
|- B e. RR
Assertion ltnegcon1i 
|- ( -u A < B <-> -u B < A )

Proof

Step Hyp Ref Expression
1 lt2.1 
 |-  A e. RR
2 lt2.2 
 |-  B e. RR
3 ltnegcon1 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )
4 1 2 3 mp2an 
 |-  ( -u A < B <-> -u B < A )