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 ∈ On → ( card ‘ 𝐴 ) ≈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		onenon	⊢ ( 𝐴 ∈ On → 𝐴 ∈ dom card )
2		cardid2	⊢ ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) ≈ 𝐴 )
3	1 2	syl	⊢ ( 𝐴 ∈ On → ( card ‘ 𝐴 ) ≈ 𝐴 )