Theorem oncardid

Description: Any ordinal number is equinumerous to its cardinal number. Unlike cardid , this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004)

	Ref	Expression
Assertion	oncardid	$$⊢A\in \mathrm{On}\to \mathrm{card}\left(A\right)\approx A$$

Proof

Step	Hyp	Ref	Expression
1		onenon	$$⊢A\in \mathrm{On}\to A\in \mathrm{dom}\mathrm{card}$$
2		cardid2	$$⊢A\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(A\right)\approx A$$
3	1 2	syl	$$⊢A\in \mathrm{On}\to \mathrm{card}\left(A\right)\approx A$$