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 ∈ On → ( card ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝑥 ≈ 𝐴 } )

Proof

Step	Hyp	Ref	Expression
1		onenon	⊢ ( 𝐴 ∈ On → 𝐴 ∈ dom card )
2		cardval3	⊢ ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝑥 ≈ 𝐴 } )
3	1 2	syl	⊢ ( 𝐴 ∈ On → ( card ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝑥 ≈ 𝐴 } )