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 ) )