Theorem oncardval

Description: The value of the cardinal number function with an ordinal number as its argument. Unlike cardval , this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003) (Revised by Mario Carneiro, 13-Sep-2013)

	Ref	Expression
Assertion	oncardval	|- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )

Proof

Step	Hyp	Ref	Expression
1		onenon	|- ( A e. On -> A e. dom card )
2		cardval3	|- ( A e. dom card -> ( card ` A ) = |^| { x e. On | x ~~ A } )
3	1 2	syl	|- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )