Theorem neirr 2258

Description: No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Assertion

Ref	Expression
neirr	⊢ ¬ 𝐴 ≠ 𝐴

Proof of Theorem neirr

Step	Hyp	Ref	Expression
1		eqid 2083	. . 3 ⊢ 𝐴 = 𝐴
2	1	notnoti 607	. 2 ⊢ ¬ ¬ 𝐴 = 𝐴
3		df-ne 2250	. . 3 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
4	3	notbii 627	. 2 ⊢ (¬ 𝐴 ≠ 𝐴 ↔ ¬ ¬ 𝐴 = 𝐴)
5	2, 4	mpbir 144	1 ⊢ ¬ 𝐴 ≠ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1285   ≠ wne 2249

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-gen 1379  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-ne 2250

This theorem is referenced by:  neldifsn  3539  0nnq  6693  1nuz2  8851