Step | Hyp | Ref
| Expression |
1 | | axpowndlem4 9422 |
. 2
⊢ (¬
∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))) |
2 | | axpowndlem1 9419 |
. . 3
⊢
(∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) |
3 | 2 | aecoms 2312 |
. 2
⊢
(∀𝑦 𝑦 = 𝑥 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) |
4 | 2 | a1d 25 |
. . 3
⊢
(∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))) |
5 | | nfnae 2318 |
. . . . . . . 8
⊢
Ⅎ𝑦 ¬
∀𝑥 𝑥 = 𝑦 |
6 | | nfae 2316 |
. . . . . . . 8
⊢
Ⅎ𝑦∀𝑦 𝑦 = 𝑧 |
7 | 5, 6 | nfan 1828 |
. . . . . . 7
⊢
Ⅎ𝑦(¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) |
8 | | el 4847 |
. . . . . . . . . . . . 13
⊢
∃𝑤 𝑥 ∈ 𝑤 |
9 | | nfcvf2 2789 |
. . . . . . . . . . . . . . 15
⊢ (¬
∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) |
10 | | nfcvd 2765 |
. . . . . . . . . . . . . . 15
⊢ (¬
∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤) |
11 | 9, 10 | nfeld 2773 |
. . . . . . . . . . . . . 14
⊢ (¬
∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑤) |
12 | | elequ2 2004 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦)) |
13 | 12 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))) |
14 | 5, 11, 13 | cbvexd 2278 |
. . . . . . . . . . . . 13
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑥 ∈ 𝑤 ↔ ∃𝑦 𝑥 ∈ 𝑦)) |
15 | 8, 14 | mpbii 223 |
. . . . . . . . . . . 12
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑦 𝑥 ∈ 𝑦) |
16 | | 19.8a 2052 |
. . . . . . . . . . . 12
⊢
(∃𝑦 𝑥 ∈ 𝑦 → ∃𝑥∃𝑦 𝑥 ∈ 𝑦) |
17 | 15, 16 | syl 17 |
. . . . . . . . . . 11
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦 𝑥 ∈ 𝑦) |
18 | | df-ex 1705 |
. . . . . . . . . . 11
⊢
(∃𝑥∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦) |
19 | 17, 18 | sylib 208 |
. . . . . . . . . 10
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦) |
20 | 19 | adantr 481 |
. . . . . . . . 9
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦) |
21 | | biidd 252 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)) |
22 | 21 | dral1 2325 |
. . . . . . . . . . . . 13
⊢
(∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ 𝑥 ∈ 𝑦 ↔ ∀𝑧 ¬ 𝑥 ∈ 𝑦)) |
23 | | alnex 1706 |
. . . . . . . . . . . . 13
⊢
(∀𝑦 ¬
𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 𝑥 ∈ 𝑦) |
24 | | alnex 1706 |
. . . . . . . . . . . . 13
⊢
(∀𝑧 ¬
𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦) |
25 | 22, 23, 24 | 3bitr3g 302 |
. . . . . . . . . . . 12
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦)) |
26 | | nd2 9410 |
. . . . . . . . . . . . 13
⊢
(∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑥 ∈ 𝑧) |
27 | | mtt 354 |
. . . . . . . . . . . . 13
⊢ (¬
∀𝑦 𝑥 ∈ 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) |
28 | 26, 27 | syl 17 |
. . . . . . . . . . . 12
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) |
29 | 25, 28 | bitrd 268 |
. . . . . . . . . . 11
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) |
30 | 29 | dral2 2324 |
. . . . . . . . . 10
⊢
(∀𝑦 𝑦 = 𝑧 → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) |
31 | 30 | adantl 482 |
. . . . . . . . 9
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))) |
32 | 20, 31 | mtbid 314 |
. . . . . . . 8
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)) |
33 | 32 | pm2.21d 118 |
. . . . . . 7
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) |
34 | 7, 33 | alrimi 2082 |
. . . . . 6
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) |
35 | | 19.8a 2052 |
. . . . . 6
⊢
(∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) |
36 | 34, 35 | syl 17 |
. . . . 5
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) |
37 | 36 | a1d 25 |
. . . 4
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) |
38 | 37 | ex 450 |
. . 3
⊢ (¬
∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))) |
39 | 4, 38 | pm2.61i 176 |
. 2
⊢
(∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) |
40 | 1, 3, 39 | pm2.61ii 177 |
1
⊢ (¬
𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) |