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Metamath Proof Explorer

Theorem ltcvrntr

Description: Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012)

Ref Expression
Hypotheses ltltncvr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
ltltncvr.s ⊢ < = ( lt ‘ 𝐾 )
ltltncvr.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
Assertion ltcvrntr ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 𝐶 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )

Proof

Step Hyp Ref Expression
1 ltltncvr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 ltltncvr.s ⊢ < = ( lt ‘ 𝐾 )
3 ltltncvr.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
4 1 2 3 cvrlt ⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑌 𝐶 𝑍 ) → 𝑌 < 𝑍 )
5 4 ex ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 𝐶 𝑍 → 𝑌 < 𝑍 ) )
6 5 3adant3r1 ⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 𝐶 𝑍 → 𝑌 < 𝑍 ) )
7 1 2 3 ltltncvr ⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )
8 6 7 sylan2d ⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 𝐶 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )