Theorem leftlt

Description: A member of a surreal's left set is less than it. (Contributed by Scott Fenton, 7-Nov-2025)

	Ref	Expression
Assertion	leftlt	⊢ ( 𝐴 ∈ ( L ‘ 𝐵 ) → 𝐴 <s 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elleft	⊢ ( 𝐴 ∈ ( L ‘ 𝐵 ) ↔ ( 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ∧ 𝐴 <s 𝐵 ) )
2	1	simprbi	⊢ ( 𝐴 ∈ ( L ‘ 𝐵 ) → 𝐴 <s 𝐵 )