Theorem pltne

Description: The "less than" relation is not reflexive. ( df-pss analog.) (Contributed by NM, 2-Dec-2011)

	Ref	Expression
Hypothesis	pltne.s	⊢ < = ( lt ‘ 𝐾 )
Assertion	pltne	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶 ) → ( 𝑋 < 𝑌 → 𝑋 ≠ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		pltne.s	⊢ < = ( lt ‘ 𝐾 )
2		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
3	2 1	pltval	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
4	3	simplbda	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝑋 < 𝑌 ) → 𝑋 ≠ 𝑌 )
5	4	ex	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶 ) → ( 𝑋 < 𝑌 → 𝑋 ≠ 𝑌 ) )