Theorem elleft

Description: Membership in the left set of a surreal. (Contributed by Scott Fenton, 7-Nov-2025)

	Ref	Expression
Assertion	elleft	⊢ ( 𝐴 ∈ ( L ‘ 𝐵 ) ↔ ( 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ∧ 𝐴 <s 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 <s 𝐵 ↔ 𝐴 <s 𝐵 ) )
2		leftval	⊢ ( L ‘ 𝐵 ) = { 𝑥 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ∣ 𝑥 <s 𝐵 }
3	1 2	elrab2	⊢ ( 𝐴 ∈ ( L ‘ 𝐵 ) ↔ ( 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ∧ 𝐴 <s 𝐵 ) )