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Metamath Proof Explorer

Theorem cardon

Description: The cardinal number of a set is an ordinal number. Proposition 10.6(1) of TakeutiZaring p. 85. (Contributed by Mario Carneiro, 7-Jan-2013) (Revised by Mario Carneiro, 13-Sep-2013)

		Ref	Expression
	Assertion	cardon	⊢ ( card ‘ 𝐴 ) ∈ On

Proof

Step	Hyp	Ref	Expression
1		cardf2	⊢ card : { 𝑥 ∣ ∃ 𝑦 ∈ On 𝑦 ≈ 𝑥 } ⟶ On
2		0elon	⊢ ∅ ∈ On
3	1 2	f0cli	⊢ ( card ‘ 𝐴 ) ∈ On