Metamath Proof Explorer


Theorem ltneg

Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by NM, 27-Aug-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion ltneg A B A < B B < A

Proof

Step Hyp Ref Expression
1 0re 0
2 ltsub2 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