Theorem cfon

Description: The cofinality of any set is an ordinal (although it only makes sense when A is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013) Avoid ax-pow and ax-un . (Revised by Umit Teoman Dogan, 10-Jun-2026)

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

Proof

Step	Hyp	Ref	Expression
1		cff	$$⊢\mathrm{cf}:\mathrm{On}⟶\mathrm{On}$$
2		0elon	$$⊢\varnothing \in \mathrm{On}$$
3	1 2	f0cli	$$⊢\mathrm{cf}\left(A\right)\in \mathrm{On}$$