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Metamath Proof Explorer

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 }