Theorem exmidne 2970

Description: Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)

Assertion

Ref	Expression
exmidne	⊢ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)

Proof of Theorem exmidne

Step	Hyp	Ref	Expression
1		neqne 2968	. 2 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)
2	1	orri 876	1 ⊢ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570   ≠ wne 2960

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-or 862  df-ne 2961

This theorem is used by:  elnn1uz2  12960  hashv01gt1  14394  numclwwlk3lem2lem  30763  fconst7v  32994  hashxpe  33181  drngmxidlr  33783  constrfin  34159  constrelextdg2  34160  subfacp1lem6  35690  tendoeq2  41581  ax6e2ndeqVD  45650  ax6e2ndeqALT  45672