Theorem nner 2340

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 2337	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2	1	biimpi 119	. 2 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
3	2	con2i 617	1 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1343   ≠ wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  nn0eln0  4597  funtpg  5239  fin0  6851  hashnncl  10709