Metamath Proof Explorer

Theorem le0neg1

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

Ref Expression
Assertion le0neg1 
|- ( A e. RR -> ( A <_ 0 <-> 0 <_ -u A ) )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 leneg 
 |-  ( ( A e. RR /\ 0 e. RR ) -> ( A <_ 0 <-> -u 0 <_ -u A ) )
3 1 2 mpan2 
 |-  ( A e. RR -> ( A <_ 0 <-> -u 0 <_ -u A ) )
4 neg0 
 |-  -u 0 = 0
5 4 breq1i 
 |-  ( -u 0 <_ -u A <-> 0 <_ -u A )
6 3 5 bitrdi 
 |-  ( A e. RR -> ( A <_ 0 <-> 0 <_ -u A ) )