Theorem neirr 2412

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 2231	. . 3 |- A = A
2	1	notnoti 650	. 2 |- -. -. A = A
3		df-ne 2404	. . 3 |- ( A =/= A <-> -. A = A )
4	3	notbii 674	. 2 |- ( -. A =/= A <-> -. -. A = A )
5	2, 4	mpbir 146	1 |- -. A =/= A

Syntax hints:   -. wn 3   = wceq 1398   =/= wne 2403

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 619  ax-in2 620  ax-gen 1498  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-ne 2404

This theorem is referenced by:  neldifsn  3807  netap  7516  2omotaplemap  7519  0nnq  7627  1nuz2  9884  umgrnloop0  16041