Theorem cvnsym 32276

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

Assertion

Ref	Expression
cvnsym	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴))

Proof of Theorem cvnsym

Step	Hyp	Ref	Expression
1		cvpss 32271	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
2		cvpss 32271	. . . . 5 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → 𝐵 ⊊ 𝐴))
3	2	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → 𝐵 ⊊ 𝐴))
4		pssn2lp 4084	. . . . 5 ⊢ ¬ (𝐵 ⊊ 𝐴 ∧ 𝐴 ⊊ 𝐵)
5	4	imnani 400	. . . 4 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊊ 𝐵)
6	3, 5	syl6 35	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → ¬ 𝐴 ⊊ 𝐵))
7	6	con2d 134	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴))
8	1, 7	syld 47	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2109   ⊊ wpss 3932   class class class wbr 5124   C_ℋ cch 30915   ⋖_ℋ ccv 30950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-cv 32265

This theorem is referenced by:  cvnref  32277