Metamath Proof Explorer


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 } )