Theorem noseqssno

Description: A surreal sequence is a subset of the surreals. (Contributed by Scott Fenton, 18-Apr-2025)

	Ref	Expression
Hypotheses	noseq.1	$$⊢\phi \to Z=\mathrm{rec}\left(\left(x\in \mathrm{V}⟼x{+}_s{1}_s\right),A\right)\left[\mathrm{\omega }\right]$$
	noseq.2	$$⊢\phi \to A\in \mathrm{No}$$
Assertion	noseqssno	$$⊢\phi \to Z\subseteq \mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		noseq.1	$$⊢\phi \to Z=\mathrm{rec}\left(\left(x\in \mathrm{V}⟼x{+}_s{1}_s\right),A\right)\left[\mathrm{\omega }\right]$$
2		noseq.2	$$⊢\phi \to A\in \mathrm{No}$$
3		peano2no	$$⊢y\in \mathrm{No}\to y{+}_s{1}_s\in \mathrm{No}$$
4	3	adantl	$$⊢\left(\phi \wedge y\in \mathrm{No}\right)\to y{+}_s{1}_s\in \mathrm{No}$$
5	1 2 2 4	noseqind	$$⊢\phi \to Z\subseteq \mathrm{No}$$