Theorem naddword2

Description: Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 15-Feb-2025)

	Ref	Expression
Assertion	naddword2	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\subseteq B+A$$

Proof

Step	Hyp	Ref	Expression
1		naddword1	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\subseteq A+B$$
2		naddcom	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A+B=B+A$$
3	1 2	sseqtrd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\subseteq B+A$$