Metamath Proof Explorer


Theorem leneg

Description: Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion leneg A B A B B A

Proof

Step Hyp Ref Expression
1 0re 0
2 lesub2 A B 0 A B 0 B 0 A
3 1 2 mp3an3 A B A B 0 B 0 A
4 df-neg B = 0 B
5 df-neg A = 0 A
6 4 5 breq12i B A 0 B 0 A
7 3 6 bitr4di A B A B B A