Theorem leftssold

Description: The left options are a subset of the old set. (Contributed by Scott Fenton, 7-Aug-2024)

	Ref	Expression
Assertion	leftssold	⊢ ( 𝑋 ∈ No → ( L ‘ 𝑋 ) ⊆ ( O ‘ ( bday ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		leftval	⊢ ( 𝑋 ∈ No → ( L ‘ 𝑋 ) = { 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑥 <s 𝑋 } )
2		ssrab2	⊢ { 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑥 <s 𝑋 } ⊆ ( O ‘ ( bday ‘ 𝑋 ) )
3	1 2	eqsstrdi	⊢ ( 𝑋 ∈ No → ( L ‘ 𝑋 ) ⊆ ( O ‘ ( bday ‘ 𝑋 ) ) )