Theorem pltnle

Description: "Less than" implies not converse "less than or equal to". (Contributed by NM, 18-Oct-2011)

	Ref	Expression
Hypotheses	pleval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pleval2.l	⊢ ≤ = ( le ‘ 𝐾 )
	pleval2.s	⊢ < = ( lt ‘ 𝐾 )
Assertion	pltnle	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ¬ 𝑌 ≤ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		pleval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		pleval2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		pleval2.s	⊢ < = ( lt ‘ 𝐾 )
4	2 3	pltval	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
5	1 2	posasymb	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) )
6	5	biimpd	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) → 𝑋 = 𝑌 ) )
7	6	expdimp	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑌 ≤ 𝑋 → 𝑋 = 𝑌 ) )
8	7	necon3ad	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ≠ 𝑌 → ¬ 𝑌 ≤ 𝑋 ) )
9	8	expimpd	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) → ¬ 𝑌 ≤ 𝑋 ) )
10	4 9	sylbid	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ¬ 𝑌 ≤ 𝑋 ) )
11	10	imp	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ¬ 𝑌 ≤ 𝑋 )