Metamath Proof Explorer

Theorem filtop

Description: The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007) (Revised by Stefan O'Rear, 28-Jul-2015)

Ref Expression
Assertion filtop 
|- ( F e. ( Fil ` X ) -> X e. F )

Proof

Step Hyp Ref Expression
1 filfbas 
 |-  ( F e. ( Fil ` X ) -> F e. ( fBas ` X ) )
2 fbasne0 
 |-  ( F e. ( fBas ` X ) -> F =/= (/) )
3 1 2 syl 
 |-  ( F e. ( Fil ` X ) -> F =/= (/) )
4 n0 
 |-  ( F =/= (/) <-> E. x x e. F )
5 filelss 
 |-  ( ( F e. ( Fil ` X ) /\ x e. F ) -> x C_ X )
6 ssid 
 |-  X C_ X
7 filss 
 |-  ( ( F e. ( Fil ` X ) /\ ( x e. F /\ X C_ X /\ x C_ X ) ) -> X e. F )
8 7 3exp2 
 |-  ( F e. ( Fil ` X ) -> ( x e. F -> ( X C_ X -> ( x C_ X -> X e. F ) ) ) )
9 8 imp 
 |-  ( ( F e. ( Fil ` X ) /\ x e. F ) -> ( X C_ X -> ( x C_ X -> X e. F ) ) )
10 6 9 mpi 
 |-  ( ( F e. ( Fil ` X ) /\ x e. F ) -> ( x C_ X -> X e. F ) )
11 5 10 mpd 
 |-  ( ( F e. ( Fil ` X ) /\ x e. F ) -> X e. F )
12 11 ex 
 |-  ( F e. ( Fil ` X ) -> ( x e. F -> X e. F ) )
13 12 exlimdv 
 |-  ( F e. ( Fil ` X ) -> ( E. x x e. F -> X e. F ) )
14 4 13 biimtrid 
 |-  ( F e. ( Fil ` X ) -> ( F =/= (/) -> X e. F ) )
15 3 14 mpd 
 |-  ( F e. ( Fil ` X ) -> X e. F )