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Theorem nner 2250
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 2247 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
21biimpi 118 . 2 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
32con2i 590 1 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1285  wne 2246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2247
This theorem is referenced by:  nn0eln0  4367  funtpg  4981  fin0  6419  sizenncl  9820
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