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Metamath Proof Explorer

Theorem cvrnbtwn

Description: There is no element between the two arguments of the covers relation. ( cvnbtwn analog.) (Contributed by NM, 18-Oct-2011)

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

Proof

Step Hyp Ref Expression
1 cvrfval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cvrfval.s ⊢ < = ( lt ‘ 𝐾 )
3 cvrfval.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
4 1 2 3 cvrval ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )
5 4 3adant3r3 ⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑌 ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )
6 ralnex ⊢ ( ∀ 𝑧 ∈ 𝐵 ¬ ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ↔ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) )
7 breq2 ⊢ ( 𝑧 = 𝑍 → ( 𝑋 < 𝑧 ↔ 𝑋 < 𝑍 ) )
8 breq1 ⊢ ( 𝑧 = 𝑍 → ( 𝑧 < 𝑌 ↔ 𝑍 < 𝑌 ) )
9 7 8 anbi12d ⊢ ( 𝑧 = 𝑍 → ( ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ↔ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
10 9 notbid ⊢ ( 𝑧 = 𝑍 → ( ¬ ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ↔ ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
11 10 rspcv ⊢ ( 𝑍 ∈ 𝐵 → ( ∀ 𝑧 ∈ 𝐵 ¬ ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
12 6 11 biimtrrid ⊢ ( 𝑍 ∈ 𝐵 → ( ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
13 12 adantld ⊢ ( 𝑍 ∈ 𝐵 → ( ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
14 13 3ad2ant3 ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
15 14 adantl ⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
16 5 15 sylbid ⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑌 → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
17 16 3impia ⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) )