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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