Theorem nonconne 2968

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 1584	. 2 ⊢ ¬ ⊥
2		eqneqall 2967	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ⊥))
3	2	imp 412	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) → ⊥)
4	1, 3	mto 200	1 ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ⊥wfal 1582   ≠ wne 2956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ne 2957

This theorem is used by:  frxp2  8154  osumcllem11N  41003  pexmidlem8N  41014  dochexmidlem8  42504