Metamath Proof Explorer


Theorem fin71num

Description: A well-orderable set is VII-finite iff it is I-finite. Thus, even without choice, on the class of well-orderable sets all eight definitions of finite set coincide. (Contributed by Mario Carneiro, 18-May-2015)

Ref Expression
Assertion fin71num
|- ( A e. dom card -> ( A e. Fin7 <-> A e. Fin ) )

Proof

Step Hyp Ref Expression
1 isfin7-2
 |-  ( A e. dom card -> ( A e. Fin7 <-> ( A e. dom card -> A e. Fin ) ) )
2 biimt
 |-  ( A e. dom card -> ( A e. Fin <-> ( A e. dom card -> A e. Fin ) ) )
3 1 2 bitr4d
 |-  ( A e. dom card -> ( A e. Fin7 <-> A e. Fin ) )