Theorem cvntr

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

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

Proof

Step	Hyp	Ref	Expression
1		cvpss	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵 ) )
2	1	3adant3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵 ) )
3		cvpss	⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐵 ⋖ℋ 𝐶 → 𝐵 ⊊ 𝐶 ) )
4	3	3adant1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐵 ⋖ℋ 𝐶 → 𝐵 ⊊ 𝐶 ) )
5		cvnbtwn	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐶 → ¬ ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) ) )
6	5	3com23	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐶 → ¬ ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) ) )
7	6	con2d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → ¬ 𝐴 ⋖ℋ 𝐶 ) )
8	2 4 7	syl2and	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ⋖ℋ 𝐵 ∧ 𝐵 ⋖ℋ 𝐶 ) → ¬ 𝐴 ⋖ℋ 𝐶 ) )