Metamath Proof Explorer


Theorem xrltned

Description: 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020)

Ref Expression
Hypotheses xrltned.1
|- ( ph -> A e. RR* )
xrltned.2
|- ( ph -> B e. RR* )
xrltned.3
|- ( ph -> A < B )
Assertion xrltned
|- ( ph -> A =/= B )

Proof

Step Hyp Ref Expression
1 xrltned.1
 |-  ( ph -> A e. RR* )
2 xrltned.2
 |-  ( ph -> B e. RR* )
3 xrltned.3
 |-  ( ph -> A < B )
4 1 2 3 xrgtned
 |-  ( ph -> B =/= A )
5 4 necomd
 |-  ( ph -> A =/= B )