Theorem bdayon

Description: The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011) (Proof shortened by Scott Fenton, 8-Dec-2021)

	Ref	Expression
Assertion	bdayon	$$⊢\mathrm{bday}\left(A\right)\in \mathrm{On}$$

Proof

Step	Hyp	Ref	Expression
1		bdayfo	$$⊢\mathrm{bday}:\mathrm{No}\underset{onto}{⟶}\mathrm{On}$$
2		fof	$$⊢\mathrm{bday}:\mathrm{No}\underset{onto}{⟶}\mathrm{On}\to \mathrm{bday}:\mathrm{No}⟶\mathrm{On}$$
3	1 2	ax-mp	$$⊢\mathrm{bday}:\mathrm{No}⟶\mathrm{On}$$
4		0elon	$$⊢\varnothing \in \mathrm{On}$$
5	3 4	f0cli	$$⊢\mathrm{bday}\left(A\right)\in \mathrm{On}$$