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 e. On -> ( card ` A ) ~~ A )

Proof

Step	Hyp	Ref	Expression
1		onenon	|- ( A e. On -> A e. dom card )
2		cardid2	|- ( A e. dom card -> ( card ` A ) ~~ A )
3	1 2	syl	|- ( A e. On -> ( card ` A ) ~~ A )