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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	$⊢ A \in On \to card (A) = ⋂ \{x \in On \| x \approx A\}$

Proof

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