Metamath Proof Explorer

Theorem lt0neg2

Description: Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004)

Ref Expression
Assertion lt0neg2 
|- ( A e. RR -> ( 0 < A <-> -u A < 0 ) )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 ltneg 
 |-  ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> -u A < -u 0 ) )
3 1 2 mpan 
 |-  ( A e. RR -> ( 0 < A <-> -u A < -u 0 ) )
4 neg0 
 |-  -u 0 = 0
5 4 breq2i 
 |-  ( -u A < -u 0 <-> -u A < 0 )
6 3 5 bitrdi 
 |-  ( A e. RR -> ( 0 < A <-> -u A < 0 ) )