Metamath Proof Explorer

Theorem undefnel2

Description: The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011)

Ref Expression
Assertion undefnel2 
|- ( S e. V -> -. ( Undef ` S ) e. S )

Proof

Step Hyp Ref Expression
1 pwuninel 
 |-  -. ~P U. S e. S
2 undefval 
 |-  ( S e. V -> ( Undef ` S ) = ~P U. S )
3 2 eleq1d 
 |-  ( S e. V -> ( ( Undef ` S ) e. S <-> ~P U. S e. S ) )
4 1 3 mtbiri 
 |-  ( S e. V -> -. ( Undef ` S ) e. S )