Theorem neirr 2315

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 2137	. . 3 |- A = A
2	1	notnoti 634	. 2 |- -. -. A = A
3		df-ne 2307	. . 3 |- ( A =/= A <-> -. A = A )
4	3	notbii 657	. 2 |- ( -. A =/= A <-> -. -. A = A )
5	2, 4	mpbir 145	1 |- -. A =/= A

Syntax hints:   -. wn 3   = wceq 1331   =/= wne 2306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-gen 1425  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-ne 2307

This theorem is referenced by:  neldifsn  3648  0nnq  7165  1nuz2  9393