Theorem nner 2286

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 2283	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2	1	biimpi 119	. 2 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
3	2	con2i 599	1 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1314   ≠ wne 2282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-in1 586  ax-in2 587

This theorem depends on definitions:  df-bi 116  df-ne 2283

This theorem is referenced by:  nn0eln0  4493  funtpg  5132  fin0  6732  hashnncl  10435