Theorem nonconne 2806

Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)

Assertion

Ref	Expression
nonconne	|- -. ( A = B /\ A =/= B )

Proof of Theorem nonconne

Step	Hyp	Ref	Expression
1		fal 1490	. 2 |- -. F.
2		eqneqall 2805	. . 3 |- ( A = B -> ( A =/= B -> F. ) )
3	2	imp 445	. 2 |- ( ( A = B /\ A =/= B ) -> F. )
4	1, 3	mto 188	1 |- -. ( A = B /\ A =/= B )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   F. wfal 1488   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-fal 1489  df-ne 2795

This theorem is referenced by:  osumcllem11N  35252  pexmidlem8N  35263  dochexmidlem8  36756