Theorem nner 2313

Description: Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)

Assertion

Ref	Expression
nner	|- ( A = B -> -. A =/= B )

Proof of Theorem nner

Step	Hyp	Ref	Expression
1		df-ne 2310	. . 3 |- ( A =/= B <-> -. A = B )
2	1	biimpi 119	. 2 |- ( A =/= B -> -. A = B )
3	2	con2i 617	1 |- ( A = B -> -. A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1332   =/= wne 2309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2310

This theorem is referenced by:  nn0eln0  4541  funtpg  5182  fin0  6787  hashnncl  10574