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)

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

Proof

Step	Hyp	Ref	Expression
1		cardcf	$$⊢\mathrm{card}\left(\mathrm{cf}\left(A\right)\right)=\mathrm{cf}\left(A\right)$$
2		cardon	$$⊢\mathrm{card}\left(\mathrm{cf}\left(A\right)\right)\in \mathrm{On}$$
3	1 2	eqeltrri	$$⊢\mathrm{cf}\left(A\right)\in \mathrm{On}$$