Theorem pltle

Description: "Less than" implies "less than or equal to". ( pssss analog.) (Contributed by NM, 4-Dec-2011)

	Ref	Expression
Hypotheses	pltval.l	⊢ ≤ = ( le ‘ 𝐾 )
	pltval.s	⊢ < = ( lt ‘ 𝐾 )
Assertion	pltle	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶 ) → ( 𝑋 < 𝑌 → 𝑋 ≤ 𝑌 ) )

Proof

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