Step | Hyp | Ref
| Expression |
1 | | r19.26 2596 |
. . . . . . 7
⊢
(∀𝑤 ∈
𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))) |
2 | | ralidm 3515 |
. . . . . . . . 9
⊢
(∀𝑤 ∈
𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤) |
3 | | r19.3rmv 3505 |
. . . . . . . . . 10
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (¬ 𝑤𝑅𝑤 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤)) |
4 | 3 | ralbidv 2470 |
. . . . . . . . 9
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤)) |
5 | 2, 4 | bitr2id 192 |
. . . . . . . 8
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤)) |
6 | 5 | anbi1d 462 |
. . . . . . 7
⊢
(∃𝑥 𝑥 ∈ 𝐴 → ((∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))) |
7 | 1, 6 | syl5bb 191 |
. . . . . 6
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))) |
8 | | r19.26 2596 |
. . . . . . 7
⊢
(∀𝑧 ∈
𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))) |
9 | 8 | ralbii 2476 |
. . . . . 6
⊢
(∀𝑤 ∈
𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))) |
10 | | r19.26 2596 |
. . . . . 6
⊢
(∀𝑤 ∈
𝐴 (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))) |
11 | 7, 9, 10 | 3bitr4g 222 |
. . . . 5
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤 ∈ 𝐴 (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))) |
12 | | r19.26 2596 |
. . . . . . . 8
⊢
(∀𝑧 ∈
𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ∧ ∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))) |
13 | | vex 2733 |
. . . . . . . . . . . . 13
⊢ 𝑧 ∈ V |
14 | 13, 13 | brcnv 4794 |
. . . . . . . . . . . 12
⊢ (𝑧◡𝑅𝑧 ↔ 𝑧𝑅𝑧) |
15 | | id 19 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤) |
16 | 15, 15 | breq12d 4002 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑤 → (𝑧𝑅𝑧 ↔ 𝑤𝑅𝑤)) |
17 | 14, 16 | syl5bb 191 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑤 → (𝑧◡𝑅𝑧 ↔ 𝑤𝑅𝑤)) |
18 | 17 | notbid 662 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑤 → (¬ 𝑧◡𝑅𝑧 ↔ ¬ 𝑤𝑅𝑤)) |
19 | 18 | cbvralv 2696 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
𝐴 ¬ 𝑧◡𝑅𝑧 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤) |
20 | | vex 2733 |
. . . . . . . . . . . . 13
⊢ 𝑦 ∈ V |
21 | 13, 20 | brcnv 4794 |
. . . . . . . . . . . 12
⊢ (𝑧◡𝑅𝑦 ↔ 𝑦𝑅𝑧) |
22 | | vex 2733 |
. . . . . . . . . . . . 13
⊢ 𝑤 ∈ V |
23 | 20, 22 | brcnv 4794 |
. . . . . . . . . . . 12
⊢ (𝑦◡𝑅𝑤 ↔ 𝑤𝑅𝑦) |
24 | 21, 23 | anbi12ci 458 |
. . . . . . . . . . 11
⊢ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) ↔ (𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧)) |
25 | 13, 22 | brcnv 4794 |
. . . . . . . . . . 11
⊢ (𝑧◡𝑅𝑤 ↔ 𝑤𝑅𝑧) |
26 | 24, 25 | imbi12i 238 |
. . . . . . . . . 10
⊢ (((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤) ↔ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) |
27 | 26 | ralbii 2476 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤) ↔ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) |
28 | 19, 27 | anbi12i 457 |
. . . . . . . 8
⊢
((∀𝑧 ∈
𝐴 ¬ 𝑧◡𝑅𝑧 ∧ ∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)) ↔ (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))) |
29 | 12, 28 | bitr2i 184 |
. . . . . . 7
⊢
((∀𝑤 ∈
𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))) |
30 | 29 | ralbii 2476 |
. . . . . 6
⊢
(∀𝑤 ∈
𝐴 (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))) |
31 | | ralcom 2633 |
. . . . . 6
⊢
(∀𝑤 ∈
𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))) |
32 | 30, 31 | bitri 183 |
. . . . 5
⊢
(∀𝑤 ∈
𝐴 (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))) |
33 | 11, 32 | bitrdi 195 |
. . . 4
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))) |
34 | 33 | ralbidv 2470 |
. . 3
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))) |
35 | | ralcom 2633 |
. . 3
⊢
(∀𝑤 ∈
𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))) |
36 | | ralcom 2633 |
. . 3
⊢
(∀𝑧 ∈
𝐴 ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))) |
37 | 34, 35, 36 | 3bitr4g 222 |
. 2
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))) |
38 | | df-po 4281 |
. 2
⊢ (𝑅 Po 𝐴 ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))) |
39 | | df-po 4281 |
. 2
⊢ (◡𝑅 Po 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))) |
40 | 37, 38, 39 | 3bitr4g 222 |
1
⊢
(∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴)) |