Theorem cvnsym 32322

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 32317	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
2		cvpss 32317	. . . . 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 4127	. . . . 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 2108   ⊊ wpss 3977   class class class wbr 5166   C_ℋ cch 30961   ⋖_ℋ ccv 30996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cv 32311

This theorem is referenced by:  cvnref  32323