Step | Hyp | Ref
| Expression |
1 | | vex 3477 |
. . . . 5
⊢ 𝑥 ∈ V |
2 | 1 | elintrab 4964 |
. . . 4
⊢ (𝑥 ∈ ∩ {𝑓
∈ (UFil‘𝑋)
∣ 𝐹 ⊆ 𝑓} ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 → 𝑥 ∈ 𝑓)) |
3 | | filsspw 23675 |
. . . . . . . . . . . . . 14
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) |
4 | 3 | 3ad2ant1 1132 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ 𝒫 𝑋) |
5 | | difss 4131 |
. . . . . . . . . . . . . . 15
⊢ (𝑋 ∖ 𝑥) ⊆ 𝑋 |
6 | | filtop 23679 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) |
7 | 6 | difexd 5329 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋 ∖ 𝑥) ∈ V) |
8 | 7 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑋 ∖ 𝑥) ∈ V) |
9 | | elpwg 4605 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∖ 𝑥) ∈ V → ((𝑋 ∖ 𝑥) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑥) ⊆ 𝑋)) |
10 | 8, 9 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑥) ⊆ 𝑋)) |
11 | 5, 10 | mpbiri 258 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑋 ∖ 𝑥) ∈ 𝒫 𝑋) |
12 | 11 | snssd 4812 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → {(𝑋 ∖ 𝑥)} ⊆ 𝒫 𝑋) |
13 | 4, 12 | unssd 4186 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ⊆ 𝒫 𝑋) |
14 | | ssun1 4172 |
. . . . . . . . . . . . . 14
⊢ 𝐹 ⊆ (𝐹 ∪ {(𝑋 ∖ 𝑥)}) |
15 | | filn0 23686 |
. . . . . . . . . . . . . 14
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅) |
16 | | ssn0 4400 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ⊆ (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ∧ 𝐹 ≠ ∅) → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ≠ ∅) |
17 | 14, 15, 16 | sylancr 586 |
. . . . . . . . . . . . 13
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ≠ ∅) |
18 | 17 | 3ad2ant1 1132 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ≠ ∅) |
19 | | elsni 4645 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ {(𝑋 ∖ 𝑥)} → 𝑧 = (𝑋 ∖ 𝑥)) |
20 | | filelss 23676 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝑋) |
21 | 20 | 3adant3 1131 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → 𝑦 ⊆ 𝑋) |
22 | | reldisj 4451 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ⊆ 𝑋 → ((𝑦 ∩ (𝑋 ∖ 𝑥)) = ∅ ↔ 𝑦 ⊆ (𝑋 ∖ (𝑋 ∖ 𝑥)))) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ((𝑦 ∩ (𝑋 ∖ 𝑥)) = ∅ ↔ 𝑦 ⊆ (𝑋 ∖ (𝑋 ∖ 𝑥)))) |
24 | | dfss4 4258 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥) |
25 | 24 | biimpi 215 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥) |
26 | 25 | sseq2d 4014 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ⊆ 𝑋 → (𝑦 ⊆ (𝑋 ∖ (𝑋 ∖ 𝑥)) ↔ 𝑦 ⊆ 𝑥)) |
27 | 26 | 3ad2ant3 1134 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑦 ⊆ (𝑋 ∖ (𝑋 ∖ 𝑥)) ↔ 𝑦 ⊆ 𝑥)) |
28 | 23, 27 | bitrd 279 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ((𝑦 ∩ (𝑋 ∖ 𝑥)) = ∅ ↔ 𝑦 ⊆ 𝑥)) |
29 | | filss 23677 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹) |
30 | 29 | 3exp2 1353 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑦 ∈ 𝐹 → (𝑥 ⊆ 𝑋 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹)))) |
31 | 30 | 3imp 1110 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹)) |
32 | 28, 31 | sylbid 239 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ((𝑦 ∩ (𝑋 ∖ 𝑥)) = ∅ → 𝑥 ∈ 𝐹)) |
33 | 32 | necon3bd 2953 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 → (𝑦 ∩ (𝑋 ∖ 𝑥)) ≠ ∅)) |
34 | 33 | 3exp 1118 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑦 ∈ 𝐹 → (𝑥 ⊆ 𝑋 → (¬ 𝑥 ∈ 𝐹 → (𝑦 ∩ (𝑋 ∖ 𝑥)) ≠ ∅)))) |
35 | 34 | com24 95 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹 ∈ (Fil‘𝑋) → (¬ 𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑋 → (𝑦 ∈ 𝐹 → (𝑦 ∩ (𝑋 ∖ 𝑥)) ≠ ∅)))) |
36 | 35 | 3imp1 1346 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐹) → (𝑦 ∩ (𝑋 ∖ 𝑥)) ≠ ∅) |
37 | | ineq2 4206 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑋 ∖ 𝑥))) |
38 | 37 | neeq1d 2999 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝑋 ∖ 𝑥) → ((𝑦 ∩ 𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑋 ∖ 𝑥)) ≠ ∅)) |
39 | 36, 38 | syl5ibrcom 246 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑦 ∈ 𝐹) → (𝑧 = (𝑋 ∖ 𝑥) → (𝑦 ∩ 𝑧) ≠ ∅)) |
40 | 39 | expimpd 453 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ((𝑦 ∈ 𝐹 ∧ 𝑧 = (𝑋 ∖ 𝑥)) → (𝑦 ∩ 𝑧) ≠ ∅)) |
41 | 19, 40 | sylan2i 605 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ((𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {(𝑋 ∖ 𝑥)}) → (𝑦 ∩ 𝑧) ≠ ∅)) |
42 | 41 | ralrimivv 3197 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {(𝑋 ∖ 𝑥)} (𝑦 ∩ 𝑧) ≠ ∅) |
43 | | filfbas 23672 |
. . . . . . . . . . . . . . 15
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) |
44 | 43 | 3ad2ant1 1132 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → 𝐹 ∈ (fBas‘𝑋)) |
45 | 5 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑋 ∖ 𝑥) ⊆ 𝑋) |
46 | 25 | 3ad2ant2 1133 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑥) = ∅) → (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥) |
47 | | difeq2 4116 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑋 ∖ 𝑥) = ∅ → (𝑋 ∖ (𝑋 ∖ 𝑥)) = (𝑋 ∖ ∅)) |
48 | | dif0 4372 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑋 ∖ ∅) = 𝑋 |
49 | 47, 48 | eqtrdi 2787 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋 ∖ 𝑥) = ∅ → (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑋) |
50 | 49 | 3ad2ant3 1134 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑥) = ∅) → (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑋) |
51 | 46, 50 | eqtr3d 2773 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑥) = ∅) → 𝑥 = 𝑋) |
52 | 6 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑥) = ∅) → 𝑋 ∈ 𝐹) |
53 | 51, 52 | eqeltrd 2832 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑥) = ∅) → 𝑥 ∈ 𝐹) |
54 | 53 | 3expia 1120 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) = ∅ → 𝑥 ∈ 𝐹)) |
55 | 54 | necon3bd 2953 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 → (𝑋 ∖ 𝑥) ≠ ∅)) |
56 | 55 | ex 412 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ⊆ 𝑋 → (¬ 𝑥 ∈ 𝐹 → (𝑋 ∖ 𝑥) ≠ ∅))) |
57 | 56 | com23 86 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ (Fil‘𝑋) → (¬ 𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑋 → (𝑋 ∖ 𝑥) ≠ ∅))) |
58 | 57 | 3imp 1110 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑋 ∖ 𝑥) ≠ ∅) |
59 | 6 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → 𝑋 ∈ 𝐹) |
60 | | snfbas 23690 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋 ∖ 𝑥) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑥) ≠ ∅ ∧ 𝑋 ∈ 𝐹) → {(𝑋 ∖ 𝑥)} ∈ (fBas‘𝑋)) |
61 | 45, 58, 59, 60 | syl3anc 1370 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → {(𝑋 ∖ 𝑥)} ∈ (fBas‘𝑋)) |
62 | | fbunfip 23693 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ {(𝑋 ∖ 𝑥)} ∈ (fBas‘𝑋)) → (¬ ∅ ∈
(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ↔ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {(𝑋 ∖ 𝑥)} (𝑦 ∩ 𝑧) ≠ ∅)) |
63 | 44, 61, 62 | syl2anc 583 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (¬ ∅ ∈
(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ↔ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {(𝑋 ∖ 𝑥)} (𝑦 ∩ 𝑧) ≠ ∅)) |
64 | 42, 63 | mpbird 257 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ¬ ∅ ∈
(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) |
65 | | fsubbas 23691 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ 𝐹 → ((fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ {(𝑋 ∖ 𝑥)}) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ≠ ∅ ∧ ¬ ∅ ∈
(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))))) |
66 | 6, 65 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝐹 ∈ (Fil‘𝑋) → ((fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ {(𝑋 ∖ 𝑥)}) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ≠ ∅ ∧ ¬ ∅ ∈
(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))))) |
67 | 66 | 3ad2ant1 1132 |
. . . . . . . . . . . 12
⊢ ((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)}))))) |
68 | 13, 18, 64, 67 | mpbir3and 1341 |
. . . . . . . . . . 11
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ∈ (fBas‘𝑋)) |
69 | | fgcl 23702 |
. . . . . . . . . . 11
⊢
((fi‘(𝐹 ∪
{(𝑋 ∖ 𝑥)})) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ∈ (Fil‘𝑋)) |
70 | 68, 69 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ∈ (Fil‘𝑋)) |
71 | | filssufil 23736 |
. . . . . . . . . . 11
⊢ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)(𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) |
72 | | snex 5431 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ {(𝑋 ∖ 𝑥)} ∈ V |
73 | | unexg 7740 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {(𝑋 ∖ 𝑥)} ∈ V) → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ∈ V) |
74 | 72, 73 | mpan2 688 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ∈ V) |
75 | | ssfii 9420 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹 ∪ {(𝑋 ∖ 𝑥)}) ∈ V → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ⊆ (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) |
76 | 74, 75 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ⊆ (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) |
77 | 76 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝐹 ∪ {(𝑋 ∖ 𝑥)}) ⊆ (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) |
78 | 77 | unssad 4187 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) |
79 | | ssfg 23696 |
. . . . . . . . . . . . . . . . . 18
⊢
((fi‘(𝐹 ∪
{(𝑋 ∖ 𝑥)})) ∈ (fBas‘𝑋) → (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})))) |
80 | 68, 79 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})))) |
81 | 78, 80 | sstrd 3992 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})))) |
82 | 81 | ad2antrr 723 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) → 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})))) |
83 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) → (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) |
84 | 82, 83 | sstrd 3992 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) → 𝐹 ⊆ 𝑓) |
85 | | ufilfil 23728 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ∈ (UFil‘𝑋) → 𝑓 ∈ (Fil‘𝑋)) |
86 | | 0nelfil 23673 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ∈ (Fil‘𝑋) → ¬ ∅ ∈
𝑓) |
87 | 85, 86 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 ∈ (UFil‘𝑋) → ¬ ∅ ∈
𝑓) |
88 | 87 | ad2antlr 724 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) → ¬ ∅ ∈ 𝑓) |
89 | | disjdif 4471 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∩ (𝑋 ∖ 𝑥)) = ∅ |
90 | 85 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋)) |
91 | | simprr 770 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → 𝑥 ∈ 𝑓) |
92 | 76 | unssbd 4188 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐹 ∈ (Fil‘𝑋) → {(𝑋 ∖ 𝑥)} ⊆ (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) |
93 | 92 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → {(𝑋 ∖ 𝑥)} ⊆ (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) |
94 | 93 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) → {(𝑋 ∖ 𝑥)} ⊆ (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) |
95 | 68 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) → (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ∈ (fBas‘𝑋)) |
96 | 95, 79 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) → (fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})))) |
97 | 94, 96 | sstrd 3992 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) → {(𝑋 ∖ 𝑥)} ⊆ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})))) |
98 | 97 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → {(𝑋 ∖ 𝑥)} ⊆ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)})))) |
99 | | simprl 768 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) |
100 | 98, 99 | sstrd 3992 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → {(𝑋 ∖ 𝑥)} ⊆ 𝑓) |
101 | | snidg 4662 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 ∖ 𝑥) ∈ V → (𝑋 ∖ 𝑥) ∈ {(𝑋 ∖ 𝑥)}) |
102 | 7, 101 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋 ∖ 𝑥) ∈ {(𝑋 ∖ 𝑥)}) |
103 | 102 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑋 ∖ 𝑥) ∈ {(𝑋 ∖ 𝑥)}) |
104 | 103 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → (𝑋 ∖ 𝑥) ∈ {(𝑋 ∖ 𝑥)}) |
105 | 100, 104 | sseldd 3983 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → (𝑋 ∖ 𝑥) ∈ 𝑓) |
106 | | filin 23678 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝑓 ∧ (𝑋 ∖ 𝑥) ∈ 𝑓) → (𝑥 ∩ (𝑋 ∖ 𝑥)) ∈ 𝑓) |
107 | 90, 91, 105, 106 | syl3anc 1370 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → (𝑥 ∩ (𝑋 ∖ 𝑥)) ∈ 𝑓) |
108 | 89, 107 | eqeltrrid 2837 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 ∧ 𝑥 ∈ 𝑓)) → ∅ ∈ 𝑓) |
109 | 108 | expr 456 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) → (𝑥 ∈ 𝑓 → ∅ ∈ 𝑓)) |
110 | 88, 109 | mtod 197 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) → ¬ 𝑥 ∈ 𝑓) |
111 | 84, 110 | jca 511 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) ∧ 𝑓 ∈ (UFil‘𝑋)) ∧ (𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓) → (𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓)) |
112 | 111 | exp31 419 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (𝑓 ∈ (UFil‘𝑋) → ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 → (𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓)))) |
113 | 112 | reximdvai 3164 |
. . . . . . . . . . 11
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → (∃𝑓 ∈ (UFil‘𝑋)(𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ⊆ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓))) |
114 | 71, 113 | syl5 34 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ((𝑋filGen(fi‘(𝐹 ∪ {(𝑋 ∖ 𝑥)}))) ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓))) |
115 | 70, 114 | mpd 15 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋) → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓)) |
116 | 115 | 3expia 1120 |
. . . . . . . 8
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹) → (𝑥 ⊆ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓))) |
117 | | filssufil 23736 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓) |
118 | | filelss 23676 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝑓) → 𝑥 ⊆ 𝑋) |
119 | 118 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝑓 → 𝑥 ⊆ 𝑋)) |
120 | 85, 119 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∈ (UFil‘𝑋) → (𝑥 ∈ 𝑓 → 𝑥 ⊆ 𝑋)) |
121 | 120 | con3d 152 |
. . . . . . . . . . . . . 14
⊢ (𝑓 ∈ (UFil‘𝑋) → (¬ 𝑥 ⊆ 𝑋 → ¬ 𝑥 ∈ 𝑓)) |
122 | 121 | impcom 407 |
. . . . . . . . . . . . 13
⊢ ((¬
𝑥 ⊆ 𝑋 ∧ 𝑓 ∈ (UFil‘𝑋)) → ¬ 𝑥 ∈ 𝑓) |
123 | 122 | a1d 25 |
. . . . . . . . . . . 12
⊢ ((¬
𝑥 ⊆ 𝑋 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐹 ⊆ 𝑓 → ¬ 𝑥 ∈ 𝑓)) |
124 | 123 | ancld 550 |
. . . . . . . . . . 11
⊢ ((¬
𝑥 ⊆ 𝑋 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐹 ⊆ 𝑓 → (𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓))) |
125 | 124 | reximdva 3167 |
. . . . . . . . . 10
⊢ (¬
𝑥 ⊆ 𝑋 → (∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓))) |
126 | 117, 125 | syl5com 31 |
. . . . . . . . 9
⊢ (𝐹 ∈ (Fil‘𝑋) → (¬ 𝑥 ⊆ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓))) |
127 | 126 | adantr 480 |
. . . . . . . 8
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹) → (¬ 𝑥 ⊆ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓))) |
128 | 116, 127 | pm2.61d 179 |
. . . . . . 7
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ¬ 𝑥 ∈ 𝐹) → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓)) |
129 | 128 | ex 412 |
. . . . . 6
⊢ (𝐹 ∈ (Fil‘𝑋) → (¬ 𝑥 ∈ 𝐹 → ∃𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓))) |
130 | | rexanali 3101 |
. . . . . 6
⊢
(∃𝑓 ∈
(UFil‘𝑋)(𝐹 ⊆ 𝑓 ∧ ¬ 𝑥 ∈ 𝑓) ↔ ¬ ∀𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 → 𝑥 ∈ 𝑓)) |
131 | 129, 130 | imbitrdi 250 |
. . . . 5
⊢ (𝐹 ∈ (Fil‘𝑋) → (¬ 𝑥 ∈ 𝐹 → ¬ ∀𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 → 𝑥 ∈ 𝑓))) |
132 | 131 | con4d 115 |
. . . 4
⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑓 ∈ (UFil‘𝑋)(𝐹 ⊆ 𝑓 → 𝑥 ∈ 𝑓) → 𝑥 ∈ 𝐹)) |
133 | 2, 132 | biimtrid 241 |
. . 3
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ ∩ {𝑓 ∈ (UFil‘𝑋) ∣ 𝐹 ⊆ 𝑓} → 𝑥 ∈ 𝐹)) |
134 | 133 | ssrdv 3988 |
. 2
⊢ (𝐹 ∈ (Fil‘𝑋) → ∩ {𝑓
∈ (UFil‘𝑋)
∣ 𝐹 ⊆ 𝑓} ⊆ 𝐹) |
135 | | ssintub 4970 |
. . 3
⊢ 𝐹 ⊆ ∩ {𝑓
∈ (UFil‘𝑋)
∣ 𝐹 ⊆ 𝑓} |
136 | 135 | a1i 11 |
. 2
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ ∩ {𝑓 ∈ (UFil‘𝑋) ∣ 𝐹 ⊆ 𝑓}) |
137 | 134, 136 | eqssd 3999 |
1
⊢ (𝐹 ∈ (Fil‘𝑋) → ∩ {𝑓
∈ (UFil‘𝑋)
∣ 𝐹 ⊆ 𝑓} = 𝐹) |