Metamath Proof Explorer


Theorem ltnled

Description: 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses ltd.1 φ A
ltd.2 φ B
Assertion ltnled φ A < B ¬ B A

Proof

Step Hyp Ref Expression
1 ltd.1 φ A
2 ltd.2 φ B
3 ltnle A B A < B ¬ B A
4 1 2 3 syl2anc φ A < B ¬ B A