Theorem rankon

Description: The rank of a set is an ordinal number. Proposition 9.15(1) of TakeutiZaring p. 79. (Contributed by NM, 5-Oct-2003) (Revised by Mario Carneiro, 12-Sep-2013)

	Ref	Expression
Assertion	rankon	⊢ ( rank ‘ 𝐴 ) ∈ On

Proof

Step	Hyp	Ref	Expression
1		rankf	⊢ rank : ∪ ( 𝑅1 “ On ) ⟶ On
2		0elon	⊢ ∅ ∈ On
3	1 2	f0cli	⊢ ( rank ‘ 𝐴 ) ∈ On