Theorem neirr 2215

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 2040	. . 3 ⊢ 𝐴 = 𝐴
2	1	notnoti 574	. 2 ⊢ ¬ ¬ 𝐴 = 𝐴
3		df-ne 2206	. . 3 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
4	3	notbii 594	. 2 ⊢ (¬ 𝐴 ≠ 𝐴 ↔ ¬ ¬ 𝐴 = 𝐴)
5	2, 4	mpbir 134	1 ⊢ ¬ 𝐴 ≠ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1243   ≠ wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-gen 1338  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-ne 2206

This theorem is referenced by:  neldifsn  3497  0nnq  6462  1nuz2  8543