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	$$⊢A\in \mathrm{On}\to \mathrm{card}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|x\approx A\right\}$$

Proof

Step	Hyp	Ref	Expression
1		onenon	$$⊢A\in \mathrm{On}\to A\in \mathrm{dom}\mathrm{card}$$
2		cardval3	$$⊢A\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|x\approx A\right\}$$
3	1 2	syl	$$⊢A\in \mathrm{On}\to \mathrm{card}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|x\approx A\right\}$$