Theorem ssltsn

Description: Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025)

	Ref	Expression
Hypotheses	ssltsn.1	$$⊢\phi \to A\in \mathrm{No}$$
	ssltsn.2	$$⊢\phi \to B\in \mathrm{No}$$
	ssltsn.3	$$⊢\phi \to A{<}_{s}B$$
Assertion	ssltsn	$$⊢\phi \to \left\{A\right\}{\ll }_s\left\{B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		ssltsn.1	$$⊢\phi \to A\in \mathrm{No}$$
2		ssltsn.2	$$⊢\phi \to B\in \mathrm{No}$$
3		ssltsn.3	$$⊢\phi \to A{<}_{s}B$$
4	1 2	ssltsnb	$$⊢\phi \to \left(\left\{A\right\}{\ll }_s\left\{B\right\}\leftrightarrow A{<}_{s}B\right)$$
5	3 4	mpbird	$$⊢\phi \to \left\{A\right\}{\ll }_s\left\{B\right\}$$