Theorem naddlid

Description: Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 20-Feb-2025)

	Ref	Expression
Assertion	naddlid	$$⊢A\in \mathrm{On}\to \varnothing +A=A$$

Proof

Step	Hyp	Ref	Expression
1		0elon	$$⊢\varnothing \in \mathrm{On}$$
2		naddcom	$$⊢\left(A\in \mathrm{On}\wedge \varnothing \in \mathrm{On}\right)\to A+\varnothing =\varnothing +A$$
3	1 2	mpan2	$$⊢A\in \mathrm{On}\to A+\varnothing =\varnothing +A$$
4		naddrid	$$⊢A\in \mathrm{On}\to A+\varnothing =A$$
5	3 4	eqtr3d	$$⊢A\in \mathrm{On}\to \varnothing +A=A$$