Metamath Proof Explorer

Theorem rankeq0

Description: A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypothesis rankeq0.1 
|- A e. _V
Assertion rankeq0 
|- ( A = (/) <-> ( rank ` A ) = (/) )

Proof

Step Hyp Ref Expression
1 rankeq0.1 
 |-  A e. _V
2 unir1 
 |-  U. ( R1 " On ) = _V
3 1 2 eleqtrri 
 |-  A e. U. ( R1 " On )
4 rankeq0b 
 |-  ( A e. U. ( R1 " On ) -> ( A = (/) <-> ( rank ` A ) = (/) ) )
5 3 4 ax-mp 
 |-  ( A = (/) <-> ( rank ` A ) = (/) )