Theorem addsge01d

Description: A surreal is less-than or equal to itself plus a non-negative surreal. (Contributed by Scott Fenton, 24-Feb-2026)

	Ref	Expression
Hypotheses	addsge01d.1	$$⊢\phi \to A\in \mathrm{No}$$
	addsge01d.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	addsge01d	$$⊢\phi \to \left(0_s{\le }_{s}B\leftrightarrow A{\le }_s\left(A{+}_{s}B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		addsge01d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsge01d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		0sno	$$⊢0_s\in \mathrm{No}$$
4	3	a1i	$$⊢\phi \to 0_s\in \mathrm{No}$$
5	4 2 1	sleadd2d	$$⊢\phi \to \left(0_s{\le }_{s}B\leftrightarrow \left(A{+}_s{0}_s\right){\le }_s\left(A{+}_{s}B\right)\right)$$
6	1	addsridd	$$⊢\phi \to A{+}_s{0}_s=A$$
7	6	breq1d	$$⊢\phi \to \left(\left(A{+}_s{0}_s\right){\le }_s\left(A{+}_{s}B\right)\leftrightarrow A{\le }_s\left(A{+}_{s}B\right)\right)$$
8	5 7	bitrd	$$⊢\phi \to \left(0_s{\le }_{s}B\leftrightarrow A{\le }_s\left(A{+}_{s}B\right)\right)$$