Theorem cvnref 32039

Description: The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
cvnref	⊢ (𝐴 ∈ Cℋ → ¬ 𝐴 ⋖ℋ 𝐴)

Proof of Theorem cvnref

Step	Hyp	Ref	Expression
1		cvnsym 32038	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐴 → ¬ 𝐴 ⋖ℋ 𝐴))
2	1	anidms 566	. 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⋖ℋ 𝐴 → ¬ 𝐴 ⋖ℋ 𝐴))
3	2	pm2.01d 189	1 ⊢ (𝐴 ∈ Cℋ → ¬ 𝐴 ⋖ℋ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2098   class class class wbr 5139   C_ℋ cch 30677   ⋖_ℋ ccv 30712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cv 32027

This theorem is referenced by: (None)