Theorem pltnlt

Description: The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011)

	Ref	Expression
Hypotheses	pltnlt.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pltnlt.s	⊢ < = ( lt ‘ 𝐾 )
Assertion	pltnlt	⊢ ( ( ( 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	3 2	pltle	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 < 𝑋 → 𝑌 ( le ‘ 𝐾 ) 𝑋 ) )
6	5	3com23	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 < 𝑋 → 𝑌 ( le ‘ 𝐾 ) 𝑋 ) )
7	6	adantr	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ( 𝑌 < 𝑋 → 𝑌 ( le ‘ 𝐾 ) 𝑋 ) )
8	4 7	mtod	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ¬ 𝑌 < 𝑋 )