Theorem nnaword2

Description: Weak ordering property of addition. (Contributed by NM, 9-Nov-2002)

	Ref	Expression
Assertion	nnaword2	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to A\subseteq B{+}_{𝑜}A$$

Proof

Step	Hyp	Ref	Expression
1		nnaword1	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to A\subseteq A{+}_{𝑜}B$$
2		nnacom	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to A{+}_{𝑜}B=B{+}_{𝑜}A$$
3	1 2	sseqtrd	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to A\subseteq B{+}_{𝑜}A$$