Theorem n0sleltp1

Description: Non-negative surreal ordering relation. (Contributed by Scott Fenton, 7-Nov-2025)

	Ref	Expression
Assertion	n0sleltp1	$$⊢\left(M\in {\mathbb{N}}_{\mathrm{0s}}\wedge N\in {\mathbb{N}}_{\mathrm{0s}}\right)\to \left(M{\le }_{s}N\leftrightarrow M{<}_s\left(N{+}_s{1}_s\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		n0sno	$$⊢M\in {\mathbb{N}}_{\mathrm{0s}}\to M\in \mathrm{No}$$
2	1	adantr	$$⊢\left(M\in {\mathbb{N}}_{\mathrm{0s}}\wedge N\in {\mathbb{N}}_{\mathrm{0s}}\right)\to M\in \mathrm{No}$$
3		n0sno	$$⊢N\in {\mathbb{N}}_{\mathrm{0s}}\to N\in \mathrm{No}$$
4	3	adantl	$$⊢\left(M\in {\mathbb{N}}_{\mathrm{0s}}\wedge N\in {\mathbb{N}}_{\mathrm{0s}}\right)\to N\in \mathrm{No}$$
5		1sno	$$⊢1_s\in \mathrm{No}$$
6	5	a1i	$$⊢\left(M\in {\mathbb{N}}_{\mathrm{0s}}\wedge N\in {\mathbb{N}}_{\mathrm{0s}}\right)\to 1_s\in \mathrm{No}$$
7	2 4 6	sleadd1d	$$⊢\left(M\in {\mathbb{N}}_{\mathrm{0s}}\wedge N\in {\mathbb{N}}_{\mathrm{0s}}\right)\to \left(M{\le }_{s}N\leftrightarrow \left(M{+}_s{1}_s\right){\le }_s\left(N{+}_s{1}_s\right)\right)$$
8		peano2n0s	$$⊢N\in {\mathbb{N}}_{\mathrm{0s}}\to N{+}_s{1}_s\in {\mathbb{N}}_{\mathrm{0s}}$$
9		n0sltp1le	$$⊢\left(M\in {\mathbb{N}}_{\mathrm{0s}}\wedge N{+}_s{1}_s\in {\mathbb{N}}_{\mathrm{0s}}\right)\to \left(M{<}_s\left(N{+}_s{1}_s\right)\leftrightarrow \left(M{+}_s{1}_s\right){\le }_s\left(N{+}_s{1}_s\right)\right)$$
10	8 9	sylan2	$$⊢\left(M\in {\mathbb{N}}_{\mathrm{0s}}\wedge N\in {\mathbb{N}}_{\mathrm{0s}}\right)\to \left(M{<}_s\left(N{+}_s{1}_s\right)\leftrightarrow \left(M{+}_s{1}_s\right){\le }_s\left(N{+}_s{1}_s\right)\right)$$
11	7 10	bitr4d	$$⊢\left(M\in {\mathbb{N}}_{\mathrm{0s}}\wedge N\in {\mathbb{N}}_{\mathrm{0s}}\right)\to \left(M{\le }_{s}N\leftrightarrow M{<}_s\left(N{+}_s{1}_s\right)\right)$$