| Step |
Hyp |
Ref |
Expression |
| 1 |
|
vex |
⊢ 𝑥 ∈ V |
| 2 |
1
|
elintrab |
⊢ ( 𝑥 ∈ ∩ { 𝑓 ∈ ( UFil ‘ 𝑋 ) ∣ 𝐹 ⊆ 𝑓 } ↔ ∀ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 → 𝑥 ∈ 𝑓 ) ) |
| 3 |
|
filsspw |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ⊆ 𝒫 𝑋 ) |
| 4 |
3
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → 𝐹 ⊆ 𝒫 𝑋 ) |
| 5 |
|
difss |
⊢ ( 𝑋 ∖ 𝑥 ) ⊆ 𝑋 |
| 6 |
|
filtop |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝑋 ∈ 𝐹 ) |
| 7 |
|
difexg |
⊢ ( 𝑋 ∈ 𝐹 → ( 𝑋 ∖ 𝑥 ) ∈ V ) |
| 8 |
6 7
|
syl |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑋 ∖ 𝑥 ) ∈ V ) |
| 9 |
8
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑋 ∖ 𝑥 ) ∈ V ) |
| 10 |
|
elpwg |
⊢ ( ( 𝑋 ∖ 𝑥 ) ∈ V → ( ( 𝑋 ∖ 𝑥 ) ∈ 𝒫 𝑋 ↔ ( 𝑋 ∖ 𝑥 ) ⊆ 𝑋 ) ) |
| 11 |
9 10
|
syl |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ( 𝑋 ∖ 𝑥 ) ∈ 𝒫 𝑋 ↔ ( 𝑋 ∖ 𝑥 ) ⊆ 𝑋 ) ) |
| 12 |
5 11
|
mpbiri |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑋 ∖ 𝑥 ) ∈ 𝒫 𝑋 ) |
| 13 |
12
|
snssd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → { ( 𝑋 ∖ 𝑥 ) } ⊆ 𝒫 𝑋 ) |
| 14 |
4 13
|
unssd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ⊆ 𝒫 𝑋 ) |
| 15 |
|
ssun1 |
⊢ 𝐹 ⊆ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) |
| 16 |
|
filn0 |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ≠ ∅ ) |
| 17 |
|
ssn0 |
⊢ ( ( 𝐹 ⊆ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ∧ 𝐹 ≠ ∅ ) → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ≠ ∅ ) |
| 18 |
15 16 17
|
sylancr |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ≠ ∅ ) |
| 19 |
18
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ≠ ∅ ) |
| 20 |
|
elsni |
⊢ ( 𝑧 ∈ { ( 𝑋 ∖ 𝑥 ) } → 𝑧 = ( 𝑋 ∖ 𝑥 ) ) |
| 21 |
|
filelss |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ) → 𝑦 ⊆ 𝑋 ) |
| 22 |
21
|
3adant3 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → 𝑦 ⊆ 𝑋 ) |
| 23 |
|
reldisj |
⊢ ( 𝑦 ⊆ 𝑋 → ( ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) = ∅ ↔ 𝑦 ⊆ ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) ) ) |
| 24 |
22 23
|
syl |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) = ∅ ↔ 𝑦 ⊆ ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) ) ) |
| 25 |
|
dfss4 |
⊢ ( 𝑥 ⊆ 𝑋 ↔ ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) = 𝑥 ) |
| 26 |
25
|
biimpi |
⊢ ( 𝑥 ⊆ 𝑋 → ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) = 𝑥 ) |
| 27 |
26
|
sseq2d |
⊢ ( 𝑥 ⊆ 𝑋 → ( 𝑦 ⊆ ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) ↔ 𝑦 ⊆ 𝑥 ) ) |
| 28 |
27
|
3ad2ant3 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑦 ⊆ ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) ↔ 𝑦 ⊆ 𝑥 ) ) |
| 29 |
24 28
|
bitrd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) = ∅ ↔ 𝑦 ⊆ 𝑥 ) ) |
| 30 |
|
filss |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ∈ 𝐹 ) |
| 31 |
30
|
3exp2 |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑦 ∈ 𝐹 → ( 𝑥 ⊆ 𝑋 → ( 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹 ) ) ) ) |
| 32 |
31
|
3imp |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹 ) ) |
| 33 |
29 32
|
sylbid |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) = ∅ → 𝑥 ∈ 𝐹 ) ) |
| 34 |
33
|
necon3bd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ¬ 𝑥 ∈ 𝐹 → ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) ≠ ∅ ) ) |
| 35 |
34
|
3exp |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑦 ∈ 𝐹 → ( 𝑥 ⊆ 𝑋 → ( ¬ 𝑥 ∈ 𝐹 → ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) ≠ ∅ ) ) ) ) |
| 36 |
35
|
com24 |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ¬ 𝑥 ∈ 𝐹 → ( 𝑥 ⊆ 𝑋 → ( 𝑦 ∈ 𝐹 → ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) ≠ ∅ ) ) ) ) |
| 37 |
36
|
3imp1 |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ) → ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) ≠ ∅ ) |
| 38 |
|
ineq2 |
⊢ ( 𝑧 = ( 𝑋 ∖ 𝑥 ) → ( 𝑦 ∩ 𝑧 ) = ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) ) |
| 39 |
38
|
neeq1d |
⊢ ( 𝑧 = ( 𝑋 ∖ 𝑥 ) → ( ( 𝑦 ∩ 𝑧 ) ≠ ∅ ↔ ( 𝑦 ∩ ( 𝑋 ∖ 𝑥 ) ) ≠ ∅ ) ) |
| 40 |
37 39
|
syl5ibrcom |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑦 ∈ 𝐹 ) → ( 𝑧 = ( 𝑋 ∖ 𝑥 ) → ( 𝑦 ∩ 𝑧 ) ≠ ∅ ) ) |
| 41 |
40
|
expimpd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ( 𝑦 ∈ 𝐹 ∧ 𝑧 = ( 𝑋 ∖ 𝑥 ) ) → ( 𝑦 ∩ 𝑧 ) ≠ ∅ ) ) |
| 42 |
20 41
|
sylan2i |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ( 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ { ( 𝑋 ∖ 𝑥 ) } ) → ( 𝑦 ∩ 𝑧 ) ≠ ∅ ) ) |
| 43 |
42
|
ralrimivv |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ∀ 𝑦 ∈ 𝐹 ∀ 𝑧 ∈ { ( 𝑋 ∖ 𝑥 ) } ( 𝑦 ∩ 𝑧 ) ≠ ∅ ) |
| 44 |
|
filfbas |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) ) |
| 45 |
44
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) ) |
| 46 |
5
|
a1i |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑋 ∖ 𝑥 ) ⊆ 𝑋 ) |
| 47 |
26
|
3ad2ant2 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ⊆ 𝑋 ∧ ( 𝑋 ∖ 𝑥 ) = ∅ ) → ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) = 𝑥 ) |
| 48 |
|
difeq2 |
⊢ ( ( 𝑋 ∖ 𝑥 ) = ∅ → ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) = ( 𝑋 ∖ ∅ ) ) |
| 49 |
|
dif0 |
⊢ ( 𝑋 ∖ ∅ ) = 𝑋 |
| 50 |
48 49
|
eqtrdi |
⊢ ( ( 𝑋 ∖ 𝑥 ) = ∅ → ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) = 𝑋 ) |
| 51 |
50
|
3ad2ant3 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ⊆ 𝑋 ∧ ( 𝑋 ∖ 𝑥 ) = ∅ ) → ( 𝑋 ∖ ( 𝑋 ∖ 𝑥 ) ) = 𝑋 ) |
| 52 |
47 51
|
eqtr3d |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ⊆ 𝑋 ∧ ( 𝑋 ∖ 𝑥 ) = ∅ ) → 𝑥 = 𝑋 ) |
| 53 |
6
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ⊆ 𝑋 ∧ ( 𝑋 ∖ 𝑥 ) = ∅ ) → 𝑋 ∈ 𝐹 ) |
| 54 |
52 53
|
eqeltrd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ⊆ 𝑋 ∧ ( 𝑋 ∖ 𝑥 ) = ∅ ) → 𝑥 ∈ 𝐹 ) |
| 55 |
54
|
3expia |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ⊆ 𝑋 ) → ( ( 𝑋 ∖ 𝑥 ) = ∅ → 𝑥 ∈ 𝐹 ) ) |
| 56 |
55
|
necon3bd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ⊆ 𝑋 ) → ( ¬ 𝑥 ∈ 𝐹 → ( 𝑋 ∖ 𝑥 ) ≠ ∅ ) ) |
| 57 |
56
|
ex |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑥 ⊆ 𝑋 → ( ¬ 𝑥 ∈ 𝐹 → ( 𝑋 ∖ 𝑥 ) ≠ ∅ ) ) ) |
| 58 |
57
|
com23 |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ¬ 𝑥 ∈ 𝐹 → ( 𝑥 ⊆ 𝑋 → ( 𝑋 ∖ 𝑥 ) ≠ ∅ ) ) ) |
| 59 |
58
|
3imp |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑋 ∖ 𝑥 ) ≠ ∅ ) |
| 60 |
6
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → 𝑋 ∈ 𝐹 ) |
| 61 |
|
snfbas |
⊢ ( ( ( 𝑋 ∖ 𝑥 ) ⊆ 𝑋 ∧ ( 𝑋 ∖ 𝑥 ) ≠ ∅ ∧ 𝑋 ∈ 𝐹 ) → { ( 𝑋 ∖ 𝑥 ) } ∈ ( fBas ‘ 𝑋 ) ) |
| 62 |
46 59 60 61
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → { ( 𝑋 ∖ 𝑥 ) } ∈ ( fBas ‘ 𝑋 ) ) |
| 63 |
|
fbunfip |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ { ( 𝑋 ∖ 𝑥 ) } ∈ ( fBas ‘ 𝑋 ) ) → ( ¬ ∅ ∈ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ↔ ∀ 𝑦 ∈ 𝐹 ∀ 𝑧 ∈ { ( 𝑋 ∖ 𝑥 ) } ( 𝑦 ∩ 𝑧 ) ≠ ∅ ) ) |
| 64 |
45 62 63
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ¬ ∅ ∈ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ↔ ∀ 𝑦 ∈ 𝐹 ∀ 𝑧 ∈ { ( 𝑋 ∖ 𝑥 ) } ( 𝑦 ∩ 𝑧 ) ≠ ∅ ) ) |
| 65 |
43 64
|
mpbird |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ¬ ∅ ∈ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) |
| 66 |
|
fsubbas |
⊢ ( 𝑋 ∈ 𝐹 → ( ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ∈ ( fBas ‘ 𝑋 ) ↔ ( ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ⊆ 𝒫 𝑋 ∧ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ≠ ∅ ∧ ¬ ∅ ∈ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) ) |
| 67 |
6 66
|
syl |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ∈ ( fBas ‘ 𝑋 ) ↔ ( ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ⊆ 𝒫 𝑋 ∧ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ≠ ∅ ∧ ¬ ∅ ∈ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) ) |
| 68 |
67
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ∈ ( fBas ‘ 𝑋 ) ↔ ( ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ⊆ 𝒫 𝑋 ∧ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ≠ ∅ ∧ ¬ ∅ ∈ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) ) |
| 69 |
14 19 65 68
|
mpbir3and |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ∈ ( fBas ‘ 𝑋 ) ) |
| 70 |
|
fgcl |
⊢ ( ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ∈ ( fBas ‘ 𝑋 ) → ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ∈ ( Fil ‘ 𝑋 ) ) |
| 71 |
69 70
|
syl |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ∈ ( Fil ‘ 𝑋 ) ) |
| 72 |
|
filssufil |
⊢ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ∈ ( Fil ‘ 𝑋 ) → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) |
| 73 |
|
snex |
⊢ { ( 𝑋 ∖ 𝑥 ) } ∈ V |
| 74 |
|
unexg |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ { ( 𝑋 ∖ 𝑥 ) } ∈ V ) → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ∈ V ) |
| 75 |
73 74
|
mpan2 |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ∈ V ) |
| 76 |
|
ssfii |
⊢ ( ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ∈ V → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ⊆ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) |
| 77 |
75 76
|
syl |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ⊆ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) |
| 78 |
77
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ⊆ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) |
| 79 |
78
|
unssad |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → 𝐹 ⊆ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) |
| 80 |
|
ssfg |
⊢ ( ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ∈ ( fBas ‘ 𝑋 ) → ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ⊆ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) |
| 81 |
69 80
|
syl |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ⊆ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) |
| 82 |
79 81
|
sstrd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → 𝐹 ⊆ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) |
| 83 |
82
|
ad2antrr |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) → 𝐹 ⊆ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) |
| 84 |
|
simpr |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) → ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) |
| 85 |
83 84
|
sstrd |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) → 𝐹 ⊆ 𝑓 ) |
| 86 |
|
ufilfil |
⊢ ( 𝑓 ∈ ( UFil ‘ 𝑋 ) → 𝑓 ∈ ( Fil ‘ 𝑋 ) ) |
| 87 |
|
0nelfil |
⊢ ( 𝑓 ∈ ( Fil ‘ 𝑋 ) → ¬ ∅ ∈ 𝑓 ) |
| 88 |
86 87
|
syl |
⊢ ( 𝑓 ∈ ( UFil ‘ 𝑋 ) → ¬ ∅ ∈ 𝑓 ) |
| 89 |
88
|
ad2antlr |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) → ¬ ∅ ∈ 𝑓 ) |
| 90 |
|
disjdif |
⊢ ( 𝑥 ∩ ( 𝑋 ∖ 𝑥 ) ) = ∅ |
| 91 |
86
|
ad2antlr |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → 𝑓 ∈ ( Fil ‘ 𝑋 ) ) |
| 92 |
|
simprr |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → 𝑥 ∈ 𝑓 ) |
| 93 |
77
|
unssbd |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → { ( 𝑋 ∖ 𝑥 ) } ⊆ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) |
| 94 |
93
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → { ( 𝑋 ∖ 𝑥 ) } ⊆ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) |
| 95 |
94
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) → { ( 𝑋 ∖ 𝑥 ) } ⊆ ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) |
| 96 |
69
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) → ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ∈ ( fBas ‘ 𝑋 ) ) |
| 97 |
96 80
|
syl |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) → ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ⊆ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) |
| 98 |
95 97
|
sstrd |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) → { ( 𝑋 ∖ 𝑥 ) } ⊆ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) |
| 99 |
98
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → { ( 𝑋 ∖ 𝑥 ) } ⊆ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ) |
| 100 |
|
simprl |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) |
| 101 |
99 100
|
sstrd |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → { ( 𝑋 ∖ 𝑥 ) } ⊆ 𝑓 ) |
| 102 |
|
snidg |
⊢ ( ( 𝑋 ∖ 𝑥 ) ∈ V → ( 𝑋 ∖ 𝑥 ) ∈ { ( 𝑋 ∖ 𝑥 ) } ) |
| 103 |
8 102
|
syl |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑋 ∖ 𝑥 ) ∈ { ( 𝑋 ∖ 𝑥 ) } ) |
| 104 |
103
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑋 ∖ 𝑥 ) ∈ { ( 𝑋 ∖ 𝑥 ) } ) |
| 105 |
104
|
ad2antrr |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → ( 𝑋 ∖ 𝑥 ) ∈ { ( 𝑋 ∖ 𝑥 ) } ) |
| 106 |
101 105
|
sseldd |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → ( 𝑋 ∖ 𝑥 ) ∈ 𝑓 ) |
| 107 |
|
filin |
⊢ ( ( 𝑓 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑓 ∧ ( 𝑋 ∖ 𝑥 ) ∈ 𝑓 ) → ( 𝑥 ∩ ( 𝑋 ∖ 𝑥 ) ) ∈ 𝑓 ) |
| 108 |
91 92 106 107
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → ( 𝑥 ∩ ( 𝑋 ∖ 𝑥 ) ) ∈ 𝑓 ) |
| 109 |
90 108
|
eqeltrrid |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓 ) ) → ∅ ∈ 𝑓 ) |
| 110 |
109
|
expr |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) → ( 𝑥 ∈ 𝑓 → ∅ ∈ 𝑓 ) ) |
| 111 |
89 110
|
mtod |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) → ¬ 𝑥 ∈ 𝑓 ) |
| 112 |
85 111
|
jca |
⊢ ( ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) ∧ ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 ) → ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) |
| 113 |
112
|
exp31 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( 𝑓 ∈ ( UFil ‘ 𝑋 ) → ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 → ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) ) |
| 114 |
113
|
reximdvai |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ⊆ 𝑓 → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) |
| 115 |
72 114
|
syl5 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ( ( 𝑋 filGen ( fi ‘ ( 𝐹 ∪ { ( 𝑋 ∖ 𝑥 ) } ) ) ) ∈ ( Fil ‘ 𝑋 ) → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) |
| 116 |
71 115
|
mpd |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ) → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) |
| 117 |
116
|
3expia |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ) → ( 𝑥 ⊆ 𝑋 → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) |
| 118 |
|
filssufil |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) 𝐹 ⊆ 𝑓 ) |
| 119 |
|
filelss |
⊢ ( ( 𝑓 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑓 ) → 𝑥 ⊆ 𝑋 ) |
| 120 |
119
|
ex |
⊢ ( 𝑓 ∈ ( Fil ‘ 𝑋 ) → ( 𝑥 ∈ 𝑓 → 𝑥 ⊆ 𝑋 ) ) |
| 121 |
86 120
|
syl |
⊢ ( 𝑓 ∈ ( UFil ‘ 𝑋 ) → ( 𝑥 ∈ 𝑓 → 𝑥 ⊆ 𝑋 ) ) |
| 122 |
121
|
con3d |
⊢ ( 𝑓 ∈ ( UFil ‘ 𝑋 ) → ( ¬ 𝑥 ⊆ 𝑋 → ¬ 𝑥 ∈ 𝑓 ) ) |
| 123 |
122
|
impcom |
⊢ ( ( ¬ 𝑥 ⊆ 𝑋 ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) → ¬ 𝑥 ∈ 𝑓 ) |
| 124 |
123
|
a1d |
⊢ ( ( ¬ 𝑥 ⊆ 𝑋 ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) → ( 𝐹 ⊆ 𝑓 → ¬ 𝑥 ∈ 𝑓 ) ) |
| 125 |
124
|
ancld |
⊢ ( ( ¬ 𝑥 ⊆ 𝑋 ∧ 𝑓 ∈ ( UFil ‘ 𝑋 ) ) → ( 𝐹 ⊆ 𝑓 → ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) |
| 126 |
125
|
reximdva |
⊢ ( ¬ 𝑥 ⊆ 𝑋 → ( ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) 𝐹 ⊆ 𝑓 → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) |
| 127 |
118 126
|
syl5com |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ¬ 𝑥 ⊆ 𝑋 → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) |
| 128 |
127
|
adantr |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ) → ( ¬ 𝑥 ⊆ 𝑋 → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) |
| 129 |
117 128
|
pm2.61d |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ¬ 𝑥 ∈ 𝐹 ) → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) |
| 130 |
129
|
ex |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ¬ 𝑥 ∈ 𝐹 → ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ) ) |
| 131 |
|
rexanali |
⊢ ( ∃ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓 ) ↔ ¬ ∀ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 → 𝑥 ∈ 𝑓 ) ) |
| 132 |
130 131
|
syl6ib |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ¬ 𝑥 ∈ 𝐹 → ¬ ∀ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 → 𝑥 ∈ 𝑓 ) ) ) |
| 133 |
132
|
con4d |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ∀ 𝑓 ∈ ( UFil ‘ 𝑋 ) ( 𝐹 ⊆ 𝑓 → 𝑥 ∈ 𝑓 ) → 𝑥 ∈ 𝐹 ) ) |
| 134 |
2 133
|
syl5bi |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑥 ∈ ∩ { 𝑓 ∈ ( UFil ‘ 𝑋 ) ∣ 𝐹 ⊆ 𝑓 } → 𝑥 ∈ 𝐹 ) ) |
| 135 |
134
|
ssrdv |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ∩ { 𝑓 ∈ ( UFil ‘ 𝑋 ) ∣ 𝐹 ⊆ 𝑓 } ⊆ 𝐹 ) |
| 136 |
|
ssintub |
⊢ 𝐹 ⊆ ∩ { 𝑓 ∈ ( UFil ‘ 𝑋 ) ∣ 𝐹 ⊆ 𝑓 } |
| 137 |
136
|
a1i |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ⊆ ∩ { 𝑓 ∈ ( UFil ‘ 𝑋 ) ∣ 𝐹 ⊆ 𝑓 } ) |
| 138 |
135 137
|
eqssd |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ∩ { 𝑓 ∈ ( UFil ‘ 𝑋 ) ∣ 𝐹 ⊆ 𝑓 } = 𝐹 ) |