Theorem neirr 2941

Description: No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Assertion

Ref	Expression
neirr	⊢ ¬ 𝐴 ≠ 𝐴

Proof of Theorem neirr

Step	Hyp	Ref	Expression
1		eqid 2736	. 2 ⊢ 𝐴 = 𝐴
2		nne 2936	. 2 ⊢ (¬ 𝐴 ≠ 𝐴 ↔ 𝐴 = 𝐴)
3	1, 2	mpbir 231	1 ⊢ ¬ 𝐴 ≠ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1542   ≠ wne 2932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2728  df-ne 2933

This theorem is referenced by:  neldifsn  4737  frxp2  8094  poxp3  8100  frxp3  8101  ac5b  10400  0nnn  12213  1nuz2  12874  dprd2da  20019  dvlog  26615  legso  28667  hleqnid  28676  umgrnloop0  29178  usgrnloop0ALT  29274  nfrgr2v  30342  0ngrp  30582  neldifpr1  32603  neldifpr2  32604  assafld  33781  signswch  34705  signstfvneq0  34716  linedegen  36325  irrdiff  37640  prtlem400  39316  padd01  40257  padd02  40258  fiiuncl  45496  gpg5nbgrvtx03starlem1  48544  gpg5nbgrvtx03starlem2  48545  gpg5nbgrvtx03starlem3  48546  gpg5nbgrvtx13starlem1  48547  gpg5nbgrvtx13starlem2  48548  gpg5nbgrvtx13starlem3  48549  gpg5edgnedg  48606  rmsupp0  48844  lcoc0  48898