This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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	$⊢ A \in On \to card (A) \approx A$

Proof

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