Metamath Proof Explorer

Theorem rankid

Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypothesis rankid.1 
|- A e. _V
Assertion rankid 
|- A e. ( R1 ` suc ( rank ` A ) )

Proof

Step Hyp Ref Expression
1 rankid.1 
 |-  A e. _V
2 unir1 
 |-  U. ( R1 " On ) = _V
3 1 2 eleqtrri 
 |-  A e. U. ( R1 " On )
4 rankidb 
 |-  ( A e. U. ( R1 " On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
5 3 4 ax-mp 
 |-  A e. ( R1 ` suc ( rank ` A ) )