Step | Hyp | Ref
| Expression |
1 | | eluni 3797 |
. . . . . 6
⊢ (𝑢 ∈ ∪ 𝐴
↔ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) |
2 | 1 | anbi2i 454 |
. . . . 5
⊢ ((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ (𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
3 | 2 | exbii 1598 |
. . . 4
⊢
(∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
4 | | 19.42v 1899 |
. . . . . . 7
⊢
(∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
5 | 4 | bicomi 131 |
. . . . . 6
⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
6 | 5 | exbii 1598 |
. . . . 5
⊢
(∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
7 | | excom 1657 |
. . . . . 6
⊢
(∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦∃𝑢(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
8 | | anass 399 |
. . . . . . . 8
⊢ (((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
9 | | ancom 264 |
. . . . . . . 8
⊢ (((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦))) |
10 | 8, 9 | bitr3i 185 |
. . . . . . 7
⊢ ((𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦))) |
11 | 10 | 2exbii 1599 |
. . . . . 6
⊢
(∃𝑦∃𝑢(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦∃𝑢(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦))) |
12 | | exdistr 1902 |
. . . . . 6
⊢
(∃𝑦∃𝑢(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦))) |
13 | 7, 11, 12 | 3bitri 205 |
. . . . 5
⊢
(∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦))) |
14 | | eluni 3797 |
. . . . . . . 8
⊢ (𝑧 ∈ ∪ 𝑦
↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) |
15 | 14 | bicomi 131 |
. . . . . . 7
⊢
(∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ↔ 𝑧 ∈ ∪ 𝑦) |
16 | 15 | anbi2i 454 |
. . . . . 6
⊢ ((𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦)) |
17 | 16 | exbii 1598 |
. . . . 5
⊢
(∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦)) |
18 | 6, 13, 17 | 3bitri 205 |
. . . 4
⊢
(∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦)) |
19 | | vex 2733 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
20 | 19 | uniex 4420 |
. . . . . . . . . 10
⊢ ∪ 𝑦
∈ V |
21 | | eleq2 2234 |
. . . . . . . . . 10
⊢ (𝑣 = ∪
𝑦 → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ ∪ 𝑦)) |
22 | 20, 21 | ceqsexv 2769 |
. . . . . . . . 9
⊢
(∃𝑣(𝑣 = ∪
𝑦 ∧ 𝑧 ∈ 𝑣) ↔ 𝑧 ∈ ∪ 𝑦) |
23 | | exancom 1601 |
. . . . . . . . 9
⊢
(∃𝑣(𝑣 = ∪
𝑦 ∧ 𝑧 ∈ 𝑣) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) |
24 | 22, 23 | bitr3i 185 |
. . . . . . . 8
⊢ (𝑧 ∈ ∪ 𝑦
↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) |
25 | 24 | anbi2i 454 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦))) |
26 | | 19.42v 1899 |
. . . . . . 7
⊢
(∃𝑣(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦))) |
27 | | ancom 264 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ ((𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦) ∧ 𝑦 ∈ 𝐴)) |
28 | | anass 399 |
. . . . . . . . 9
⊢ (((𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
29 | 27, 28 | bitri 183 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ (𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
30 | 29 | exbii 1598 |
. . . . . . 7
⊢
(∃𝑣(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
31 | 25, 26, 30 | 3bitr2i 207 |
. . . . . 6
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
32 | 31 | exbii 1598 |
. . . . 5
⊢
(∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑦∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
33 | | excom 1657 |
. . . . 5
⊢
(∃𝑦∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
34 | | exdistr 1902 |
. . . . . 6
⊢
(∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
35 | | vex 2733 |
. . . . . . . . . 10
⊢ 𝑣 ∈ V |
36 | | eqeq1 2177 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑣 → (𝑥 = ∪ 𝑦 ↔ 𝑣 = ∪ 𝑦)) |
37 | 36 | anbi1d 462 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑣 → ((𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
38 | 37 | exbidv 1818 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑣 → (∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))) |
39 | 35, 38 | elab 2874 |
. . . . . . . . 9
⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} ↔ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) |
40 | 39 | bicomi 131 |
. . . . . . . 8
⊢
(∃𝑦(𝑣 = ∪
𝑦 ∧ 𝑦 ∈ 𝐴) ↔ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}) |
41 | 40 | anbi2i 454 |
. . . . . . 7
⊢ ((𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})) |
42 | 41 | exbii 1598 |
. . . . . 6
⊢
(∃𝑣(𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})) |
43 | 34, 42 | bitri 183 |
. . . . 5
⊢
(∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})) |
44 | 32, 33, 43 | 3bitri 205 |
. . . 4
⊢
(∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})) |
45 | 3, 18, 44 | 3bitri 205 |
. . 3
⊢
(∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})) |
46 | 45 | abbii 2286 |
. 2
⊢ {𝑧 ∣ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴)} = {𝑧 ∣ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})} |
47 | | df-uni 3795 |
. 2
⊢ ∪ ∪ 𝐴 = {𝑧 ∣ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴)} |
48 | | df-uni 3795 |
. 2
⊢ ∪ {𝑥
∣ ∃𝑦(𝑥 = ∪
𝑦 ∧ 𝑦 ∈ 𝐴)} = {𝑧 ∣ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})} |
49 | 46, 47, 48 | 3eqtr4i 2201 |
1
⊢ ∪ ∪ 𝐴 = ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} |