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Metamath Proof Explorer

Theorem pltn2lp

Description: The less-than relation has no 2-cycle loops. ( pssn2lp analog.) (Contributed by NM, 2-Dec-2011)

Ref Expression
Hypotheses pltnlt.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
pltnlt.s ⊢ < = ( lt ‘ 𝐾 )
Assertion pltn2lp ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ¬ ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑋 ) )

Proof

Step Hyp Ref Expression
1 pltnlt.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 pltnlt.s ⊢ < = ( lt ‘ 𝐾 )
3 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
4 1 3 2 pltnle ⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ¬ 𝑌 ( le ‘ 𝐾 ) 𝑋 )
5 4 ex ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ¬ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) )
6 3 2 pltle ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 < 𝑋 → 𝑌 ( le ‘ 𝐾 ) 𝑋 ) )
7 6 3com23 ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 < 𝑋 → 𝑌 ( le ‘ 𝐾 ) 𝑋 ) )
8 5 7 nsyld ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ¬ 𝑌 < 𝑋 ) )
9 imnan ⊢ ( ( 𝑋 < 𝑌 → ¬ 𝑌 < 𝑋 ) ↔ ¬ ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑋 ) )
10 8 9 sylib ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ¬ ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑋 ) )