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  4621  funtpg  5269  fin0  6888  hashnncl  10778