Step | Hyp | Ref
| Expression |
1 | | snssi 4339 |
. . . . . 6
⊢ (𝑝 ∈ 𝑆 → {𝑝} ⊆ 𝑆) |
2 | | neiss 20913 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ {𝑝} ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘{𝑝})) |
3 | 1, 2 | syl3an3 1361 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑝 ∈ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘{𝑝})) |
4 | 3 | 3exp 1264 |
. . . 4
⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (𝑝 ∈ 𝑆 → 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))) |
5 | 4 | ralrimdv 2968 |
. . 3
⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))) |
6 | 5 | 3ad2ant1 1082 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))) |
7 | | r19.28zv 4066 |
. . . . 5
⊢ (𝑆 ≠ ∅ →
(∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) ↔ (𝑁 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
8 | 7 | 3ad2ant3 1084 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) ↔ (𝑁 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
9 | | ssrab2 3687 |
. . . . . . . . . 10
⊢ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝐽 |
10 | | uniopn 20702 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝐽) → ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽) |
11 | 9, 10 | mpan2 707 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → ∪ {𝑣
∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽) |
12 | 11 | ad2antrr 762 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → ∪
{𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽) |
13 | | sseq1 3626 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = 𝑔 → (𝑣 ⊆ 𝑁 ↔ 𝑔 ⊆ 𝑁)) |
14 | 13 | elrab 3363 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ (𝑔 ∈ 𝐽 ∧ 𝑔 ⊆ 𝑁)) |
15 | | elunii 4441 |
. . . . . . . . . . . . . . 15
⊢ ((𝑝 ∈ 𝑔 ∧ 𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) |
16 | 14, 15 | sylan2br 493 |
. . . . . . . . . . . . . 14
⊢ ((𝑝 ∈ 𝑔 ∧ (𝑔 ∈ 𝐽 ∧ 𝑔 ⊆ 𝑁)) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) |
17 | 16 | an12s 843 |
. . . . . . . . . . . . 13
⊢ ((𝑔 ∈ 𝐽 ∧ (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) |
18 | 17 | rexlimiva 3028 |
. . . . . . . . . . . 12
⊢
(∃𝑔 ∈
𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) |
19 | 18 | ralimi 2952 |
. . . . . . . . . . 11
⊢
(∀𝑝 ∈
𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∀𝑝 ∈ 𝑆 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) |
20 | | dfss3 3592 |
. . . . . . . . . . 11
⊢ (𝑆 ⊆ ∪ {𝑣
∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ ∀𝑝 ∈ 𝑆 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) |
21 | 19, 20 | sylibr 224 |
. . . . . . . . . 10
⊢
(∀𝑝 ∈
𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) |
22 | 21 | adantl 482 |
. . . . . . . . 9
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) |
23 | | unissb 4469 |
. . . . . . . . . 10
⊢ (∪ {𝑣
∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁 ↔ ∀ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}ℎ ⊆ 𝑁) |
24 | | sseq1 3626 |
. . . . . . . . . . . 12
⊢ (𝑣 = ℎ → (𝑣 ⊆ 𝑁 ↔ ℎ ⊆ 𝑁)) |
25 | 24 | elrab 3363 |
. . . . . . . . . . 11
⊢ (ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ (ℎ ∈ 𝐽 ∧ ℎ ⊆ 𝑁)) |
26 | 25 | simprbi 480 |
. . . . . . . . . 10
⊢ (ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → ℎ ⊆ 𝑁) |
27 | 23, 26 | mprgbir 2927 |
. . . . . . . . 9
⊢ ∪ {𝑣
∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁 |
28 | 22, 27 | jctir 561 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∧ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁)) |
29 | | sseq2 3627 |
. . . . . . . . . 10
⊢ (ℎ = ∪
{𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → (𝑆 ⊆ ℎ ↔ 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})) |
30 | | sseq1 3626 |
. . . . . . . . . 10
⊢ (ℎ = ∪
{𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → (ℎ ⊆ 𝑁 ↔ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁)) |
31 | 29, 30 | anbi12d 747 |
. . . . . . . . 9
⊢ (ℎ = ∪
{𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → ((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ↔ (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∧ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁))) |
32 | 31 | rspcev 3309 |
. . . . . . . 8
⊢ ((∪ {𝑣
∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽 ∧ (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∧ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)) |
33 | 12, 28, 32 | syl2anc 693 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)) |
34 | 33 | ex 450 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))) |
35 | 34 | anim2d 589 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑁 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)))) |
36 | 35 | 3adant3 1081 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((𝑁 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)))) |
37 | 8, 36 | sylbid 230 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)))) |
38 | | ssel2 3598 |
. . . . . . 7
⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑆) → 𝑝 ∈ 𝑋) |
39 | | neips.1 |
. . . . . . . 8
⊢ 𝑋 = ∪
𝐽 |
40 | 39 | isneip 20909 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
41 | 38, 40 | sylan2 491 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑆)) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
42 | 41 | anassrs 680 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑆) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
43 | 42 | ralbidva 2985 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ ∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
44 | 43 | 3adant3 1081 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ ∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
45 | 39 | isnei 20907 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)))) |
46 | 45 | 3adant3 1081 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)))) |
47 | 37, 44, 46 | 3imtr4d 283 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
48 | 6, 47 | impbid 202 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))) |