Metamath Proof Explorer


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