Theorem elleft

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

	Ref	Expression
Assertion	elleft	$$⊢A\in L\left(B\right)\leftrightarrow \left(A\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)\wedge A{<}_{s}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq1	$$⊢x=A\to \left(x{<}_{s}B\leftrightarrow A{<}_{s}B\right)$$
2		leftval	$$⊢L\left(B\right)=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)|x{<}_{s}B\right\}$$
3	1 2	elrab2	$$⊢A\in L\left(B\right)\leftrightarrow \left(A\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)\wedge A{<}_{s}B\right)$$