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Theorem nner 2210
Description: Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
Assertion
Ref Expression
nner (𝐴 = 𝐵 → ¬ 𝐴𝐵)

Proof of Theorem nner
StepHypRef Expression
1 df-ne 2206 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
21biimpi 113 . 2 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
32con2i 557 1 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1243  wne 2204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110  df-ne 2206
This theorem is referenced by:  nn0eln0  4328  funtpg  4937  fin0  6329
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