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	⊢ ( cf ‘ 𝐴 ) ∈ On

Proof

Step	Hyp	Ref	Expression
1		cardcf	⊢ ( card ‘ ( cf ‘ 𝐴 ) ) = ( cf ‘ 𝐴 )
2		cardon	⊢ ( card ‘ ( cf ‘ 𝐴 ) ) ∈ On
3	1 2	eqeltrri	⊢ ( cf ‘ 𝐴 ) ∈ On