Theorem leftssold

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

	Ref	Expression
Assertion	leftssold	$$⊢X\in \mathrm{No}\to L\left(X\right)\subseteq \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		leftval	$$⊢X\in \mathrm{No}\to L\left(X\right)=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)|x{<}_{s}X\right\}$$
2		ssrab2	$$⊢\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)|x{<}_{s}X\right\}\subseteq \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)$$
3	1 2	eqsstrdi	$$⊢X\in \mathrm{No}\to L\left(X\right)\subseteq \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)$$