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 } ) |
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 } ) |