Metamath Proof Explorer


Theorem ltne

Description: 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999) (Revised by Mario Carneiro, 16-Sep-2015)

Ref Expression
Assertion ltne
|- ( ( A e. RR /\ A < B ) -> B =/= A )

Proof

Step Hyp Ref Expression
1 ltnr
 |-  ( A e. RR -> -. A < A )
2 breq2
 |-  ( B = A -> ( A < B <-> A < A ) )
3 2 notbid
 |-  ( B = A -> ( -. A < B <-> -. A < A ) )
4 1 3 syl5ibrcom
 |-  ( A e. RR -> ( B = A -> -. A < B ) )
5 4 necon2ad
 |-  ( A e. RR -> ( A < B -> B =/= A ) )
6 5 imp
 |-  ( ( A e. RR /\ A < B ) -> B =/= A )