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