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 ∈ On → ( card ‘ 𝐴 ) = { 𝑥 ∈ On ∣ 𝑥𝐴 } )

Proof

Step Hyp Ref Expression
1 onenon ( 𝐴 ∈ On → 𝐴 ∈ dom card )
2 cardval3 ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) = { 𝑥 ∈ On ∣ 𝑥𝐴 } )
3 1 2 syl ( 𝐴 ∈ On → ( card ‘ 𝐴 ) = { 𝑥 ∈ On ∣ 𝑥𝐴 } )