Theorem nner 2351

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 2348	. . 3 |- ( A =/= B <-> -. A = B )
2	1	biimpi 120	. 2 |- ( A =/= B -> -. A = B )
3	2	con2i 627	1 |- ( A = B -> -. A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   =/= wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-ne 2348

This theorem is referenced by:  nn0eln0  4619  funtpg  5267  fin0  6884  hashnncl  10774