Theorem nner 2259

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 2256	. . 3 |- ( A =/= B <-> -. A = B )
2	1	biimpi 118	. 2 |- ( A =/= B -> -. A = B )
3	2	con2i 592	1 |- ( A = B -> -. A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1289   =/= wne 2255

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-in1 579  ax-in2 580

This theorem depends on definitions:  df-bi 115  df-ne 2256

This theorem is referenced by:  nn0eln0  4433  funtpg  5065  fin0  6601  hashnncl  10204