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	$$⊢\phi \to A\in L\left(B\right)$$
Assertion	leftoldd	$$⊢\phi \to A\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		leftel.1	$$⊢\phi \to A\in L\left(B\right)$$
2		leftold	$$⊢A\in L\left(B\right)\to A\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)$$
3	1 2	syl	$$⊢\phi \to A\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)$$