Metamath Proof Explorer


Theorem ltltncvr

Description: A chained strong ordering is not a 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 ltltncvr ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 < 𝑌𝑌 < 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 simpll ( ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝐾𝐴 )
5 simplr1 ( ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝑋𝐵 )
6 simplr3 ( ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝑍𝐵 )
7 simplr2 ( ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝑌𝐵 )
8 simpr ( ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝑋 𝐶 𝑍 )
9 1 2 3 cvrnbtwn ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑍𝐵𝑌𝐵 ) ∧ 𝑋 𝐶 𝑍 ) → ¬ ( 𝑋 < 𝑌𝑌 < 𝑍 ) )
10 4 5 6 7 8 9 syl131anc ( ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → ¬ ( 𝑋 < 𝑌𝑌 < 𝑍 ) )
11 10 ex ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑋 𝐶 𝑍 → ¬ ( 𝑋 < 𝑌𝑌 < 𝑍 ) ) )
12 11 con2d ( ( 𝐾𝐴 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 < 𝑌𝑌 < 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )