This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nonconne

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

		Ref	Expression
	Assertion	nonconne	$⊢ \neg (A = B \land A \neq B)$

Step	Hyp	Ref	Expression
1		fal	$⊢ \neg ⊥$
2		eqneqall	$⊢ A = B \to (A \neq B \to ⊥)$
3	2	imp	$⊢ (A = B \land A \neq B) \to ⊥$
4	1 3	mto	$⊢ \neg (A = B \land A \neq B)$