Metamath Proof Explorer

Theorem nonconne

Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012) (Proof shortened by Wolf Lammen, 21-Dec-2019)

Ref Expression
Assertion nonconne 
|- -. ( A = B /\ A =/= B )

Proof

Step Hyp Ref Expression
1 fal 
 |-  -. F.
2 eqneqall 
 |-  ( A = B -> ( A =/= B -> F. ) )
3 2 imp 
 |-  ( ( A = B /\ A =/= B ) -> F. )
4 1 3 mto 
 |-  -. ( A = B /\ A =/= B )