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 ) )