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	⊢ ¬ ( 𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		fal	⊢ ¬ ⊥
2		eqneqall	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≠ 𝐵 → ⊥ ) )
3	2	imp	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵 ) → ⊥ )
4	1 3	mto	⊢ ¬ ( 𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵 )