Theorem cvnref 30074

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 30073	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐴 → ¬ 𝐴 ⋖ℋ 𝐴))
2	1	anidms 570	. 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⋖ℋ 𝐴 → ¬ 𝐴 ⋖ℋ 𝐴))
3	2	pm2.01d 193	1 ⊢ (𝐴 ∈ Cℋ → ¬ 𝐴 ⋖ℋ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111   class class class wbr 5030   C_ℋ cch 28712   ⋖_ℋ ccv 28747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-cv 30062

This theorem is referenced by: (None)