Theorem leftoldd

Description: An element of a left set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026)

	Ref	Expression
Hypothesis	leftel.1	⊢ ( 𝜑 → 𝐴 ∈ ( L ‘ 𝐵 ) )
Assertion	leftoldd	⊢ ( 𝜑 → 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		leftel.1	⊢ ( 𝜑 → 𝐴 ∈ ( L ‘ 𝐵 ) )
2		leftold	⊢ ( 𝐴 ∈ ( L ‘ 𝐵 ) → 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) )
3	1 2	syl	⊢ ( 𝜑 → 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) )