Metamath Proof Explorer

Theorem gtned

Description: 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses ltd.1 
|- ( ph -> A e. RR )
ltned.2 
|- ( ph -> A < B )
Assertion gtned 
|- ( ph -> B =/= A )

Proof

Step Hyp Ref Expression
1 ltd.1 
 |-  ( ph -> A e. RR )
2 ltned.2 
 |-  ( ph -> A < B )
3 ltne 
 |-  ( ( A e. RR /\ A < B ) -> B =/= A )
4 1 2 3 syl2anc 
 |-  ( ph -> B =/= A )