Theorem nonconne 2999

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 1552	. 2 ⊢ ¬ ⊥
2		eqneqall 2998	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ⊥))
3	2	imp 410	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) → ⊥)
4	1, 3	mto 200	1 ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538  ⊥wfal 1550   ≠ wne 2987

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 210  df-an 400  df-tru 1541  df-fal 1551  df-ne 2988

This theorem is referenced by:  osumcllem11N  37262  pexmidlem8N  37273  dochexmidlem8  38763