Step | Hyp | Ref
| Expression |
1 | | inss2 3796 |
. . . . . . . . . 10
⊢ (𝑗 ∩ 𝐴) ⊆ 𝐴 |
2 | | vex 3176 |
. . . . . . . . . . . 12
⊢ 𝑗 ∈ V |
3 | 2 | inex1 4727 |
. . . . . . . . . . 11
⊢ (𝑗 ∩ 𝐴) ∈ V |
4 | 3 | elpw 4114 |
. . . . . . . . . 10
⊢ ((𝑗 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑗 ∩ 𝐴) ⊆ 𝐴) |
5 | 1, 4 | mpbir 220 |
. . . . . . . . 9
⊢ (𝑗 ∩ 𝐴) ∈ 𝒫 𝐴 |
6 | | eleq1 2676 |
. . . . . . . . 9
⊢ (𝑎 = (𝑗 ∩ 𝐴) → (𝑎 ∈ 𝒫 𝐴 ↔ (𝑗 ∩ 𝐴) ∈ 𝒫 𝐴)) |
7 | 5, 6 | mpbiri 247 |
. . . . . . . 8
⊢ (𝑎 = (𝑗 ∩ 𝐴) → 𝑎 ∈ 𝒫 𝐴) |
8 | 7 | adantl 481 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) → 𝑎 ∈ 𝒫 𝐴) |
9 | 8 | rexlimivw 3011 |
. . . . . 6
⊢
(∃𝑗 ∈
𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) → 𝑎 ∈ 𝒫 𝐴) |
10 | 9 | abssi 3640 |
. . . . 5
⊢ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ⊆ 𝒫 𝐴 |
11 | | haustop 20945 |
. . . . . . . . 9
⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) |
12 | | hauspwpwf1.x |
. . . . . . . . . 10
⊢ 𝑋 = ∪
𝐽 |
13 | 12 | topopn 20536 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
14 | 11, 13 | syl 17 |
. . . . . . . 8
⊢ (𝐽 ∈ Haus → 𝑋 ∈ 𝐽) |
15 | | ssexg 4732 |
. . . . . . . 8
⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V) |
16 | 14, 15 | sylan2 490 |
. . . . . . 7
⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐽 ∈ Haus) → 𝐴 ∈ V) |
17 | 16 | ancoms 468 |
. . . . . 6
⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
18 | | pwexg 4776 |
. . . . . 6
⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) |
19 | | elpw2g 4754 |
. . . . . 6
⊢
(𝒫 𝐴 ∈
V → ({𝑎 ∣
∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∈ 𝒫 𝒫 𝐴 ↔ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ⊆ 𝒫 𝐴)) |
20 | 17, 18, 19 | 3syl 18 |
. . . . 5
⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∈ 𝒫 𝒫 𝐴 ↔ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ⊆ 𝒫 𝐴)) |
21 | 10, 20 | mpbiri 247 |
. . . 4
⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∈ 𝒫 𝒫 𝐴) |
22 | 21 | a1d 25 |
. . 3
⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∈ 𝒫 𝒫 𝐴)) |
23 | | simplll 794 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝐽 ∈ Haus) |
24 | 12 | clsss3 20673 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋) |
25 | 11, 24 | sylan 487 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋) |
26 | 25 | ad2antrr 758 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋) |
27 | | simplrl 796 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ ((cls‘𝐽)‘𝐴)) |
28 | 26, 27 | sseldd 3569 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ 𝑋) |
29 | | simplrr 797 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ ((cls‘𝐽)‘𝐴)) |
30 | 26, 29 | sseldd 3569 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ 𝑋) |
31 | | simpr 476 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦) |
32 | 12 | hausnei 20942 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ≠ 𝑦)) → ∃𝑘 ∈ 𝐽 ∃𝑙 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) |
33 | 23, 28, 30, 31, 32 | syl13anc 1320 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → ∃𝑘 ∈ 𝐽 ∃𝑙 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) |
34 | | simprll 798 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → 𝑘 ∈ 𝐽) |
35 | | simprr1 1102 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → 𝑥 ∈ 𝑘) |
36 | | eqidd 2611 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴)) |
37 | | elequ2 1991 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑘 → (𝑥 ∈ 𝑗 ↔ 𝑥 ∈ 𝑘)) |
38 | | ineq1 3769 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑘 → (𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴)) |
39 | 38 | eqeq2d 2620 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑘 → ((𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴))) |
40 | 37, 39 | anbi12d 743 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑘 → ((𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑘 ∧ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴)))) |
41 | 40 | rspcev 3282 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ 𝐽 ∧ (𝑥 ∈ 𝑘 ∧ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴))) → ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
42 | 34, 35, 36, 41 | syl12anc 1316 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
43 | | vex 3176 |
. . . . . . . . . . . . . 14
⊢ 𝑘 ∈ V |
44 | 43 | inex1 4727 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∩ 𝐴) ∈ V |
45 | | eqeq1 2614 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = (𝑘 ∩ 𝐴) → (𝑎 = (𝑗 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
46 | 45 | anbi2d 736 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑘 ∩ 𝐴) → ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))) |
47 | 46 | rexbidv 3034 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑘 ∩ 𝐴) → (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))) |
48 | 44, 47 | elab 3319 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
49 | 42, 48 | sylibr 223 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → (𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) |
50 | 11 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top) |
51 | 50 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝐽 ∈ Top) |
52 | | simplr 788 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐴 ⊆ 𝑋) |
53 | 52 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝐴 ⊆ 𝑋) |
54 | | simprr 792 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝑦 ∈ ((cls‘𝐽)‘𝐴)) |
55 | 54 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑦 ∈ ((cls‘𝐽)‘𝐴)) |
56 | | simplr 788 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → 𝑙 ∈ 𝐽) |
57 | 56 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑙 ∈ 𝐽) |
58 | | simprl 790 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑗 ∈ 𝐽) |
59 | | inopn 20529 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐽 ∈ Top ∧ 𝑙 ∈ 𝐽 ∧ 𝑗 ∈ 𝐽) → (𝑙 ∩ 𝑗) ∈ 𝐽) |
60 | 51, 57, 58, 59 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → (𝑙 ∩ 𝑗) ∈ 𝐽) |
61 | | simpr2 1061 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → 𝑦 ∈ 𝑙) |
62 | 61 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑦 ∈ 𝑙) |
63 | | simprr 792 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑦 ∈ 𝑗) |
64 | 62, 63 | elind 3760 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑦 ∈ (𝑙 ∩ 𝑗)) |
65 | 12 | clsndisj 20689 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴)) ∧ ((𝑙 ∩ 𝑗) ∈ 𝐽 ∧ 𝑦 ∈ (𝑙 ∩ 𝑗))) → ((𝑙 ∩ 𝑗) ∩ 𝐴) ≠ ∅) |
66 | 51, 53, 55, 60, 64, 65 | syl32anc 1326 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → ((𝑙 ∩ 𝑗) ∩ 𝐴) ≠ ∅) |
67 | | n0 3890 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑙 ∩ 𝑗) ∩ 𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴)) |
68 | 66, 67 | sylib 207 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → ∃𝑧 𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴)) |
69 | | elin 3758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) ↔ (𝑧 ∈ (𝑙 ∩ 𝑗) ∧ 𝑧 ∈ 𝐴)) |
70 | | elin 3758 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ (𝑙 ∩ 𝑗) ↔ (𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗)) |
71 | 70 | anbi1i 727 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑧 ∈ (𝑙 ∩ 𝑗) ∧ 𝑧 ∈ 𝐴) ↔ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴)) |
72 | 69, 71 | bitri 263 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) ↔ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴)) |
73 | | elin 3758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 ∈ (𝑗 ∩ 𝐴) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ∈ 𝐴)) |
74 | 73 | biimpri 217 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑧 ∈ 𝑗 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝑗 ∩ 𝐴)) |
75 | 74 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝑗 ∩ 𝐴)) |
76 | 75 | ad2antll 761 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → 𝑧 ∈ (𝑗 ∩ 𝐴)) |
77 | | simpll 786 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝑙) |
78 | 77 | ad2antll 761 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → 𝑧 ∈ 𝑙) |
79 | | simpr3 1062 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → (𝑘 ∩ 𝑙) = ∅) |
80 | 79 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → (𝑘 ∩ 𝑙) = ∅) |
81 | | minel 3985 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑧 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → ¬ 𝑧 ∈ 𝑘) |
82 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∩ 𝐴) ⊆ 𝑘 |
83 | 82 | sseli 3564 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 ∈ (𝑘 ∩ 𝐴) → 𝑧 ∈ 𝑘) |
84 | 81, 83 | nsyl 134 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑧 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → ¬ 𝑧 ∈ (𝑘 ∩ 𝐴)) |
85 | 78, 80, 84 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → ¬ 𝑧 ∈ (𝑘 ∩ 𝐴)) |
86 | | nelneq2 2713 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑧 ∈ (𝑗 ∩ 𝐴) ∧ ¬ 𝑧 ∈ (𝑘 ∩ 𝐴)) → ¬ (𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴)) |
87 | 76, 85, 86 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → ¬ (𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴)) |
88 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)) |
89 | 87, 88 | sylnib 317 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)) |
90 | 89 | expr 641 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → (((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
91 | 72, 90 | syl5bi 231 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → (𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
92 | 91 | exlimdv 1848 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → (∃𝑧 𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
93 | 68, 92 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)) |
94 | 93 | anassrs 678 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ 𝑗 ∈ 𝐽) ∧ 𝑦 ∈ 𝑗) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)) |
95 | | nan 602 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ 𝑗 ∈ 𝐽) → ¬ (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) ↔ (((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ 𝑗 ∈ 𝐽) ∧ 𝑦 ∈ 𝑗) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
96 | 94, 95 | mpbir 220 |
. . . . . . . . . . . . 13
⊢
((((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ 𝑗 ∈ 𝐽) → ¬ (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
97 | 96 | nrexdv 2984 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → ¬ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
98 | 45 | anbi2d 736 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑘 ∩ 𝐴) → ((𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))) |
99 | 98 | rexbidv 3034 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑘 ∩ 𝐴) → (∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))) |
100 | 44, 99 | elab 3319 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) |
101 | 97, 100 | sylnibr 318 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → ¬ (𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) |
102 | | nelne1 2878 |
. . . . . . . . . . 11
⊢ (((𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∧ ¬ (𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) |
103 | 49, 101, 102 | syl2anc 691 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) |
104 | 103 | expr 641 |
. . . . . . . . 9
⊢
(((((𝐽 ∈ Haus
∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ (𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽)) → ((𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})) |
105 | 104 | rexlimdvva 3020 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → (∃𝑘 ∈ 𝐽 ∃𝑙 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})) |
106 | 33, 105 | mpd 15 |
. . . . . . 7
⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) |
107 | 106 | ex 449 |
. . . . . 6
⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → (𝑥 ≠ 𝑦 → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})) |
108 | 107 | necon4d 2806 |
. . . . 5
⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} → 𝑥 = 𝑦)) |
109 | | eleq1 2676 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑗 ↔ 𝑦 ∈ 𝑗)) |
110 | 109 | anbi1d 737 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)))) |
111 | 110 | rexbidv 3034 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)))) |
112 | 111 | abbidv 2728 |
. . . . 5
⊢ (𝑥 = 𝑦 → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) |
113 | 108, 112 | impbid1 214 |
. . . 4
⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ 𝑥 = 𝑦)) |
114 | 113 | ex 449 |
. . 3
⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴)) → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ 𝑥 = 𝑦))) |
115 | 22, 114 | dom2lem 7881 |
. 2
⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴) |
116 | | hauspwpwf1.f |
. . 3
⊢ 𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) |
117 | | f1eq1 6009 |
. . 3
⊢ (𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) → (𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴 ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)) |
118 | 116, 117 | ax-mp 5 |
. 2
⊢ (𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴 ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴) |
119 | 115, 118 | sylibr 223 |
1
⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴) |