Step | Hyp | Ref
| Expression |
1 | | ishaus2 21527 |
. 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))) |
2 | | topontop 21089 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
3 | | simp1 1172 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝐽 ∈ Top) |
4 | 3 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝐽 ∈ Top) |
5 | | simp2 1173 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑚 ∈ 𝐽) |
6 | 5 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑚 ∈ 𝐽) |
7 | | simp1 1172 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑥 ∈ 𝑚) |
8 | 7 | adantl 475 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑥 ∈ 𝑚) |
9 | | opnneip 21295 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑥 ∈ 𝑚) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥})) |
10 | 4, 6, 8, 9 | syl3anc 1496 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥})) |
11 | | simp3 1174 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑛 ∈ 𝐽) |
12 | 11 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑛 ∈ 𝐽) |
13 | | simp2 1173 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑦 ∈ 𝑛) |
14 | 13 | adantl 475 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑦 ∈ 𝑛) |
15 | | opnneip 21295 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽 ∧ 𝑦 ∈ 𝑛) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦})) |
16 | 4, 12, 14, 15 | syl3anc 1496 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦})) |
17 | | simp3 1174 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → (𝑚 ∩ 𝑛) = ∅) |
18 | 17 | adantl 475 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → (𝑚 ∩ 𝑛) = ∅) |
19 | | ineq1 4035 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑚 → (𝑢 ∩ 𝑣) = (𝑚 ∩ 𝑣)) |
20 | 19 | eqeq1d 2828 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑚 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑣) = ∅)) |
21 | | ineq2 4036 |
. . . . . . . . . . . 12
⊢ (𝑣 = 𝑛 → (𝑚 ∩ 𝑣) = (𝑚 ∩ 𝑛)) |
22 | 21 | eqeq1d 2828 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑛 → ((𝑚 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑛) = ∅)) |
23 | 20, 22 | rspc2ev 3542 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ ((nei‘𝐽)‘{𝑥}) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑦}) ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅) |
24 | 10, 16, 18, 23 | syl3anc 1496 |
. . . . . . . . 9
⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅) |
25 | 24 | ex 403 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)) |
26 | 25 | 3expib 1158 |
. . . . . . 7
⊢ (𝐽 ∈ Top → ((𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))) |
27 | 26 | rexlimdvv 3248 |
. . . . . 6
⊢ (𝐽 ∈ Top → (∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)) |
28 | | neii2 21284 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢)) |
29 | 28 | ex 403 |
. . . . . . . 8
⊢ (𝐽 ∈ Top → (𝑢 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))) |
30 | | neii2 21284 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑦})) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) |
31 | 30 | ex 403 |
. . . . . . . 8
⊢ (𝐽 ∈ Top → (𝑣 ∈ ((nei‘𝐽)‘{𝑦}) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))) |
32 | | vex 3418 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ∈ V |
33 | 32 | snss 4536 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝑚 ↔ {𝑥} ⊆ 𝑚) |
34 | 33 | anbi1i 619 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ↔ ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢)) |
35 | | vex 3418 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑦 ∈ V |
36 | 35 | snss 4536 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ 𝑛 ↔ {𝑦} ⊆ 𝑛) |
37 | 36 | anbi1i 619 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) |
38 | | simp1l 1260 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑥 ∈ 𝑚) |
39 | | simp2l 1262 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑦 ∈ 𝑛) |
40 | | ss2in 4066 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → (𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣)) |
41 | | ssn0 4202 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) ∧ (𝑚 ∩ 𝑛) ≠ ∅) → (𝑢 ∩ 𝑣) ≠ ∅) |
42 | 41 | ex 403 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑚 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ 𝑣) ≠ ∅)) |
43 | 42 | necon4d 3024 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅)) |
44 | 40, 43 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅)) |
45 | 44 | ad2ant2l 754 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅)) |
46 | 45 | 3impia 1151 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑚 ∩ 𝑛) = ∅) |
47 | 38, 39, 46 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) |
48 | 47 | 3exp 1154 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))) |
49 | 37, 48 | syl5bir 235 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))) |
50 | 49 | com3r 87 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑢 ∩ 𝑣) = ∅ → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))) |
51 | 50 | imp 397 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) |
52 | 51 | 3adant1 1166 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) |
53 | 52 | reximdv 3225 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) |
54 | 53 | 3exp 1154 |
. . . . . . . . . . . . . . 15
⊢ (𝐽 ∈ Top → ((𝑢 ∩ 𝑣) = ∅ → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))) |
55 | 54 | com34 91 |
. . . . . . . . . . . . . 14
⊢ (𝐽 ∈ Top → ((𝑢 ∩ 𝑣) = ∅ → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))) |
56 | 55 | 3imp 1143 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) |
57 | 34, 56 | syl5bir 235 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → (({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) |
58 | 57 | reximdv 3225 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → (∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) |
59 | 58 | 3exp 1154 |
. . . . . . . . . 10
⊢ (𝐽 ∈ Top → ((𝑢 ∩ 𝑣) = ∅ → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))) |
60 | 59 | com24 95 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → (∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))) |
61 | 60 | impd 400 |
. . . . . . . 8
⊢ (𝐽 ∈ Top →
((∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))) |
62 | 29, 31, 61 | syl2and 603 |
. . . . . . 7
⊢ (𝐽 ∈ Top → ((𝑢 ∈ ((nei‘𝐽)‘{𝑥}) ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑦})) → ((𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))) |
63 | 62 | rexlimdvv 3248 |
. . . . . 6
⊢ (𝐽 ∈ Top → (∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) |
64 | 27, 63 | impbid 204 |
. . . . 5
⊢ (𝐽 ∈ Top → (∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)) |
65 | 64 | imbi2d 332 |
. . . 4
⊢ (𝐽 ∈ Top → ((𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))) |
66 | 65 | 2ralbidv 3199 |
. . 3
⊢ (𝐽 ∈ Top →
(∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))) |
67 | 2, 66 | syl 17 |
. 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))) |
68 | 1, 67 | bitrd 271 |
1
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))) |