Step | Hyp | Ref
| Expression |
1 | | elirrv 9591 |
. . . . . . . 8
⊢ ¬
𝑥 ∈ 𝑥 |
2 | | pm5.501 367 |
. . . . . . . 8
⊢ (¬
𝑥 ∈ 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))) |
3 | 1, 2 | mp1i 13 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))) |
4 | | elequ1 2114 |
. . . . . . . . 9
⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑥 ↔ 𝑥 ∈ 𝑥)) |
5 | 4 | notbid 318 |
. . . . . . . 8
⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥)) |
6 | | elequ1 2114 |
. . . . . . . . 9
⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) |
7 | 6 | rexbidv 3179 |
. . . . . . . 8
⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
8 | 5, 7 | bibi12d 346 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → ((¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))) |
9 | 3, 8 | bitr4d 282 |
. . . . . 6
⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))) |
10 | 9 | biimpd 228 |
. . . . 5
⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))) |
11 | 10 | spimevw 1999 |
. . . 4
⊢
(∃𝑦 ∈
𝐴 𝑥 ∈ 𝑦 → ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
12 | | ssel 3976 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On)) |
13 | 12 | adantr 482 |
. . . . . . . . . 10
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐴 → 𝑦 ∈ On)) |
14 | 13 | imp 408 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On) |
15 | | ssel2 3978 |
. . . . . . . . . 10
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) |
16 | 15 | adantr 482 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On) |
17 | | ontri1 6399 |
. . . . . . . . 9
⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦)) |
18 | 14, 16, 17 | syl2anc 585 |
. . . . . . . 8
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦)) |
19 | 18 | ralbidva 3176 |
. . . . . . 7
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)) |
20 | | ralnex 3073 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
21 | 19, 20 | bitrdi 287 |
. . . . . 6
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
22 | | unissb 4944 |
. . . . . . . . . 10
⊢ (∪ 𝐴
⊆ 𝑥 ↔
∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) |
23 | | simpr 486 |
. . . . . . . . . . 11
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 ⊆ 𝑥) |
24 | | elssuni 4942 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) |
25 | 24 | ad2antlr 726 |
. . . . . . . . . . 11
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → 𝑥 ⊆ ∪ 𝐴) |
26 | 23, 25 | eqssd 4000 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 = 𝑥) |
27 | 22, 26 | sylan2br 596 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ∪ 𝐴 = 𝑥) |
28 | | dfuni2 4911 |
. . . . . . . . . . 11
⊢ ∪ 𝐴 =
{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} |
29 | 28 | eqeq1i 2738 |
. . . . . . . . . 10
⊢ (∪ 𝐴 =
𝑥 ↔ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} = 𝑥) |
30 | | eqabcb 2876 |
. . . . . . . . . 10
⊢ ({𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} = 𝑥 ↔ ∀𝑧(∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥)) |
31 | | bicom 221 |
. . . . . . . . . . 11
⊢
((∃𝑦 ∈
𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
32 | 31 | albii 1822 |
. . . . . . . . . 10
⊢
(∀𝑧(∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
33 | 29, 30, 32 | 3bitri 297 |
. . . . . . . . 9
⊢ (∪ 𝐴 =
𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
34 | 27, 33 | sylib 217 |
. . . . . . . 8
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
35 | | notnotb 315 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ 𝑥 ↔ ¬ ¬ 𝑧 ∈ 𝑥) |
36 | 35 | bibi1i 339 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ ¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
37 | | nbbn 385 |
. . . . . . . . . . 11
⊢ ((¬
¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
38 | 36, 37 | bitri 275 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
39 | 38 | albii 1822 |
. . . . . . . . 9
⊢
(∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑧 ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
40 | | alnex 1784 |
. . . . . . . . 9
⊢
(∀𝑧 ¬
(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
41 | 39, 40 | bitri 275 |
. . . . . . . 8
⊢
(∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
42 | 34, 41 | sylib 217 |
. . . . . . 7
⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
43 | 42 | ex 414 |
. . . . . 6
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))) |
44 | 21, 43 | sylbird 260 |
. . . . 5
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))) |
45 | 44 | con4d 115 |
. . . 4
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
46 | 11, 45 | impbid2 225 |
. . 3
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))) |
47 | 46 | ralbidva 3176 |
. 2
⊢ (𝐴 ⊆ On →
(∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))) |
48 | | dminxp 6180 |
. . 3
⊢ (dom ( E
∩ (𝐴 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 E 𝑦) |
49 | | epel 5584 |
. . . . 5
⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) |
50 | 49 | rexbii 3095 |
. . . 4
⊢
(∃𝑦 ∈
𝐴 𝑥 E 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
51 | 50 | ralbii 3094 |
. . 3
⊢
(∀𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐴 𝑥 E 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
52 | 48, 51 | bitri 275 |
. 2
⊢ (dom ( E
∩ (𝐴 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
53 | | ralnex 3073 |
. . . 4
⊢
(∀𝑥 ∈
𝐴 ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
54 | | exnal 1830 |
. . . . . 6
⊢
(∃𝑧 ¬
(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
55 | | nbbn 385 |
. . . . . . . 8
⊢ ((¬
𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
56 | 55 | bicomi 223 |
. . . . . . 7
⊢ (¬
(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
57 | 56 | exbii 1851 |
. . . . . 6
⊢
(∃𝑧 ¬
(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
58 | 54, 57 | bitr3i 277 |
. . . . 5
⊢ (¬
∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
59 | 58 | ralbii 3094 |
. . . 4
⊢
(∀𝑥 ∈
𝐴 ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
60 | 53, 59 | bitr3i 277 |
. . 3
⊢ (¬
∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
61 | | uniel 41966 |
. . 3
⊢ (∪ 𝐴
∈ 𝐴 ↔
∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
62 | 60, 61 | xchnxbir 333 |
. 2
⊢ (¬
∪ 𝐴 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) |
63 | 47, 52, 62 | 3bitr4g 314 |
1
⊢ (𝐴 ⊆ On → (dom ( E
∩ (𝐴 × 𝐴)) = 𝐴 ↔ ¬ ∪
𝐴 ∈ 𝐴)) |