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Metamath Proof Explorer

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 ) )