Theorem neirr 2389

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 2209	. . 3 ⊢ 𝐴 = 𝐴
2	1	notnoti 648	. 2 ⊢ ¬ ¬ 𝐴 = 𝐴
3		df-ne 2381	. . 3 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
4	3	notbii 672	. 2 ⊢ (¬ 𝐴 ≠ 𝐴 ↔ ¬ ¬ 𝐴 = 𝐴)
5	2, 4	mpbir 146	1 ⊢ ¬ 𝐴 ≠ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1375   ≠ wne 2380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-gen 1475  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-cleq 2202  df-ne 2381

This theorem is referenced by:  neldifsn  3777  netap  7408  2omotaplemap  7411  0nnq  7519  1nuz2  9769  umgrnloop0  15882