Theorem nnasuc

Description: Addition with successor. Theorem 4I(A2) of Enderton p. 79. (Contributed by NM, 20-Sep-1995) (Revised by Mario Carneiro, 14-Nov-2014)

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

Proof

Step	Hyp	Ref	Expression
1		nnon	$$⊢A\in \mathrm{\omega }\to A\in \mathrm{On}$$
2		onasuc	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{\omega }\right)\to A{+}_{𝑜}\mathrm{suc}B=\mathrm{suc}\left(A{+}_{𝑜}B\right)$$
3	1 2	sylan	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to A{+}_{𝑜}\mathrm{suc}B=\mathrm{suc}\left(A{+}_{𝑜}B\right)$$