Theorem rankval3

Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of TakeutiZaring p. 79. (Contributed by NM, 11-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankval3.1	⊢ 𝐴 ∈ V
Assertion	rankval3	⊢ ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑦 ) ∈ 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		rankval3.1	⊢ 𝐴 ∈ V
2		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
3	1 2	eleqtrri	⊢ 𝐴 ∈ ∪ ( 𝑅1 “ On )
4		rankval3b	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑦 ) ∈ 𝑥 } )
5	3 4	ax-mp	⊢ ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑦 ) ∈ 𝑥 }