Theorem neirr 2429

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 2238	. . 3 |- A = A
2	1	notnoti 654	. 2 |- -. -. A = A
3		df-ne 2421	. . 3 |- ( A =/= A <-> -. A = A )
4	3	notbii 678	. 2 |- ( -. A =/= A <-> -. -. A = A )
5	2, 4	mpbir 146	1 |- -. A =/= A

This proof depends on syntax axioms:   -. wn 3   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-gen 1502  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is used by:  neldifsn  3844  netap  7620  2omotaplemap  7623  0nnq  7731  1nuz2  10006  umgrnloop0  16358