Theorem neirr 2229

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 2056	. . 3 ⊢ 𝐴 = 𝐴
2	1	notnoti 584	. 2 ⊢ ¬ ¬ 𝐴 = 𝐴
3		df-ne 2221	. . 3 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
4	3	notbii 604	. 2 ⊢ (¬ 𝐴 ≠ 𝐴 ↔ ¬ ¬ 𝐴 = 𝐴)
5	2, 4	mpbir 138	1 ⊢ ¬ 𝐴 ≠ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1259   ≠ wne 2220

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-gen 1354  ax-ext 2038

This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-ne 2221

This theorem is referenced by:  neldifsn  3524  0nnq  6519  1nuz2  8639