Theorem cvnsym

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

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

Proof

Step	Hyp	Ref	Expression
1		cvpss	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵 ) )
2		cvpss	⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( 𝐵 ⋖ℋ 𝐴 → 𝐵 ⊊ 𝐴 ) )
3	2	ancoms	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⋖ℋ 𝐴 → 𝐵 ⊊ 𝐴 ) )
4		pssn2lp	⊢ ¬ ( 𝐵 ⊊ 𝐴 ∧ 𝐴 ⊊ 𝐵 )
5	4	imnani	⊢ ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊊ 𝐵 )
6	3 5	syl6	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⋖ℋ 𝐴 → ¬ 𝐴 ⊊ 𝐵 ) )
7	6	con2d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊊ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴 ) )
8	1 7	syld	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴 ) )