Theorem elons

Description: Membership in the class of surreal ordinals. (Contributed by Scott Fenton, 18-Mar-2025)

	Ref	Expression
Assertion	elons	$$⊢A\in {On}_s\leftrightarrow \left(A\in \mathrm{No}\wedge R\left(A\right)=\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		fveq2	$$⊢x=A\to R\left(x\right)=R\left(A\right)$$
2	1	eqeq1d	$$⊢x=A\to \left(R\left(x\right)=\varnothing \leftrightarrow R\left(A\right)=\varnothing \right)$$
3		df-ons	$$⊢{On}_s=\left\{x\in \mathrm{No}|R\left(x\right)=\varnothing \right\}$$
4	2 3	elrab2	$$⊢A\in {On}_s\leftrightarrow \left(A\in \mathrm{No}\wedge R\left(A\right)=\varnothing \right)$$