Theorem onsled

Description: Less-than or equal is the same as non-strict birthday comparison over surreal ordinals. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026)

	Ref	Expression
Hypotheses	onsltd.1	$$⊢\phi \to A\in {On}_s$$
	onsltd.2	$$⊢\phi \to B\in {On}_s$$
Assertion	onsled	$$⊢\phi \to \left(A{\le }_{s}B\leftrightarrow \mathrm{bday}\left(A\right)\subseteq \mathrm{bday}\left(B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		onsltd.1	$$⊢\phi \to A\in {On}_s$$
2		onsltd.2	$$⊢\phi \to B\in {On}_s$$
3		onsle	$$⊢\left(A\in {On}_s\wedge B\in {On}_s\right)\to \left(A{\le }_{s}B\leftrightarrow \mathrm{bday}\left(A\right)\subseteq \mathrm{bday}\left(B\right)\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A{\le }_{s}B\leftrightarrow \mathrm{bday}\left(A\right)\subseteq \mathrm{bday}\left(B\right)\right)$$