Theorem cvnsym 32551

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 32546	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
2		cvpss 32546	. . . . 5 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → 𝐵 ⊊ 𝐴))
3	2	ancoms 463	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → 𝐵 ⊊ 𝐴))
4		pssn2lp 4061	. . . . 5 ⊢ ¬ (𝐵 ⊊ 𝐴 ∧ 𝐴 ⊊ 𝐵)
5	4	imnani 405	. . . 4 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊊ 𝐵)
6	3, 5	syl6 36	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐴 → ¬ 𝐴 ⊊ 𝐵))
7	6	con2d 135	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴))
8	1, 7	syld 48	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2145   ⊊ wpss 3908   class class class wbr 5105   C_ℋ cch 31190   ⋖_ℋ ccv 31225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-cv 32540

This theorem is referenced by:  cvnref  32552