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	$$⊢\neg \left(A=B\wedge A\ne B\right)$$

Proof

Step	Hyp	Ref	Expression
1		fal	$$⊢\neg \perp$$
2		eqneqall	$$⊢A=B\to \left(A\ne B\to \perp \right)$$
3	2	imp	$$⊢\left(A=B\wedge A\ne B\right)\to \perp$$
4	1 3	mto	$$⊢\neg \left(A=B\wedge A\ne B\right)$$