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 FFilXXF

Proof

Step Hyp Ref Expression
1 filfbas FFilXFfBasX
2 fbasne0 FfBasXF
3 1 2 syl FFilXF
4 n0 FxxF
5 filelss FFilXxFxX
6 ssid XX
7 filss FFilXxFXXxXXF
8 7 3exp2 FFilXxFXXxXXF
9 8 imp FFilXxFXXxXXF
10 6 9 mpi FFilXxFxXXF
11 5 10 mpd FFilXxFXF
12 11 ex FFilXxFXF
13 12 exlimdv FFilXxxFXF
14 4 13 biimtrid FFilXFXF
15 3 14 mpd FFilXXF