Step | Hyp | Ref
| Expression |
1 | | r1funlim 9280 |
. . . . 5
⊢ (Fun
𝑅1 ∧ Lim dom 𝑅1) |
2 | 1 | simpri 489 |
. . . 4
⊢ Lim dom
𝑅1 |
3 | | limord 6241 |
. . . 4
⊢ (Lim dom
𝑅1 → Ord dom 𝑅1) |
4 | 2, 3 | ax-mp 5 |
. . 3
⊢ Ord dom
𝑅1 |
5 | | ordelon 6206 |
. . 3
⊢ ((Ord dom
𝑅1 ∧ 𝐴 ∈ dom 𝑅1) →
𝐴 ∈
On) |
6 | 4, 5 | mpan 690 |
. 2
⊢ (𝐴 ∈ dom
𝑅1 → 𝐴 ∈ On) |
7 | | eleq1 2821 |
. . . 4
⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔
𝑦 ∈ dom
𝑅1)) |
8 | | eleq1 2821 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪
(𝑅1 “ On) ↔ 𝑦 ∈ ∪
(𝑅1 “ On))) |
9 | | fveq2 6686 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦)) |
10 | | id 22 |
. . . . . 6
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) |
11 | 9, 10 | eqeq12d 2755 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦)) |
12 | 8, 11 | anbi12d 634 |
. . . 4
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))) |
13 | 7, 12 | imbi12d 348 |
. . 3
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 →
(𝑥 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)))) |
14 | | eleq1 2821 |
. . . 4
⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1 ↔
𝐴 ∈ dom
𝑅1)) |
15 | | eleq1 2821 |
. . . . 5
⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪
(𝑅1 “ On) ↔ 𝐴 ∈ ∪
(𝑅1 “ On))) |
16 | | fveq2 6686 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴)) |
17 | | id 22 |
. . . . . 6
⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) |
18 | 16, 17 | eqeq12d 2755 |
. . . . 5
⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴)) |
19 | 15, 18 | anbi12d 634 |
. . . 4
⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))) |
20 | 14, 19 | imbi12d 348 |
. . 3
⊢ (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅1 →
(𝑥 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅1 →
(𝐴 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝐴) = 𝐴)))) |
21 | | ordtr1 6225 |
. . . . . . . . . 10
⊢ (Ord dom
𝑅1 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) →
𝑦 ∈ dom
𝑅1)) |
22 | 4, 21 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) →
𝑦 ∈ dom
𝑅1) |
23 | 22 | ancoms 462 |
. . . . . . . 8
⊢ ((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom
𝑅1) |
24 | | pm5.5 365 |
. . . . . . . 8
⊢ (𝑦 ∈ dom
𝑅1 → ((𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))) |
25 | 23, 24 | syl 17 |
. . . . . . 7
⊢ ((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))) |
26 | 25 | ralbidva 3109 |
. . . . . 6
⊢ (𝑥 ∈ dom
𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))) |
27 | | simplr 769 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥) |
28 | | ordelon 6206 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Ord dom
𝑅1 ∧ 𝑥 ∈ dom 𝑅1) →
𝑥 ∈
On) |
29 | 4, 28 | mpan 690 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ dom
𝑅1 → 𝑥 ∈ On) |
30 | 29 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On) |
31 | | eloni 6192 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ On → Ord 𝑥) |
32 | 30, 31 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → Ord 𝑥) |
33 | | ordelsuc 7566 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥)) |
34 | 27, 32, 33 | syl2anc 587 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥)) |
35 | 27, 34 | mpbid 235 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ⊆ 𝑥) |
36 | 23 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ dom
𝑅1) |
37 | | limsuc 7595 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (Lim dom
𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc
𝑦 ∈ dom
𝑅1)) |
38 | 2, 37 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ dom
𝑅1 ↔ suc 𝑦 ∈ dom
𝑅1) |
39 | 36, 38 | sylib 221 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom
𝑅1) |
40 | | simpll 767 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom
𝑅1) |
41 | | r1ord3g 9293 |
. . . . . . . . . . . . . . . . . 18
⊢ ((suc
𝑦 ∈ dom
𝑅1 ∧ 𝑥 ∈ dom 𝑅1) →
(suc 𝑦 ⊆ 𝑥 →
(𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥))) |
42 | 39, 40, 41 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc
𝑦) ⊆
(𝑅1‘𝑥))) |
43 | 35, 42 | mpd 15 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc
𝑦) ⊆
(𝑅1‘𝑥)) |
44 | | rankidb 9314 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈
(𝑅1‘suc (rank‘𝑦))) |
45 | 44 | ad2antrl 728 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc
(rank‘𝑦))) |
46 | | suceq 6247 |
. . . . . . . . . . . . . . . . . . 19
⊢
((rank‘𝑦) =
𝑦 → suc
(rank‘𝑦) = suc 𝑦) |
47 | 46 | ad2antll 729 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦) |
48 | 47 | fveq2d 6690 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc
(rank‘𝑦)) =
(𝑅1‘suc 𝑦)) |
49 | 45, 48 | eleqtrd 2836 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc
𝑦)) |
50 | 43, 49 | sseldd 3888 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘𝑥)) |
51 | 50 | ex 416 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅1‘𝑥))) |
52 | 51 | ralimdva 3092 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ dom
𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))) |
53 | 52 | imp 410 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥)) |
54 | | dfss3 3875 |
. . . . . . . . . . . 12
⊢ (𝑥 ⊆
(𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥)) |
55 | 53, 54 | sylibr 237 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅1‘𝑥)) |
56 | | vex 3404 |
. . . . . . . . . . . 12
⊢ 𝑥 ∈ V |
57 | 56 | elpw 4502 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝒫
(𝑅1‘𝑥) ↔ 𝑥 ⊆ (𝑅1‘𝑥)) |
58 | 55, 57 | sylibr 237 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫
(𝑅1‘𝑥)) |
59 | | r1sucg 9283 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ dom
𝑅1 → (𝑅1‘suc 𝑥) = 𝒫
(𝑅1‘𝑥)) |
60 | 59 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc
𝑥) = 𝒫
(𝑅1‘𝑥)) |
61 | 58, 60 | eleqtrrd 2837 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc
𝑥)) |
62 | | r1elwf 9310 |
. . . . . . . . 9
⊢ (𝑥 ∈
(𝑅1‘suc 𝑥) → 𝑥 ∈ ∪
(𝑅1 “ On)) |
63 | 61, 62 | syl 17 |
. . . . . . . 8
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ ∪
(𝑅1 “ On)) |
64 | | rankval3b 9340 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ∪ (𝑅1 “ On) →
(rank‘𝑥) = ∩ {𝑧
∈ On ∣ ∀𝑦
∈ 𝑥 (rank‘𝑦) ∈ 𝑧}) |
65 | 63, 64 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧}) |
66 | | eleq1 2821 |
. . . . . . . . . . . . . . . 16
⊢
((rank‘𝑦) =
𝑦 → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
67 | 66 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
68 | 67 | ralimi 3076 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
69 | | ralbi 3083 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 ∈
𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)) |
70 | 68, 69 | syl 17 |
. . . . . . . . . . . . 13
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)) |
71 | | dfss3 3875 |
. . . . . . . . . . . . 13
⊢ (𝑥 ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧) |
72 | 70, 71 | bitr4di 292 |
. . . . . . . . . . . 12
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ 𝑥 ⊆ 𝑧)) |
73 | 72 | rabbidv 3382 |
. . . . . . . . . . 11
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧}) |
74 | 73 | inteqd 4851 |
. . . . . . . . . 10
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧}) |
75 | 74 | adantl 485 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧}) |
76 | 29 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On) |
77 | | intmin 4866 |
. . . . . . . . . 10
⊢ (𝑥 ∈ On → ∩ {𝑧
∈ On ∣ 𝑥 ⊆
𝑧} = 𝑥) |
78 | 76, 77 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥) |
79 | 65, 75, 78 | 3eqtrd 2778 |
. . . . . . . 8
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥) |
80 | 63, 79 | jca 515 |
. . . . . . 7
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) |
81 | 80 | ex 416 |
. . . . . 6
⊢ (𝑥 ∈ dom
𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))) |
82 | 26, 81 | sylbid 243 |
. . . . 5
⊢ (𝑥 ∈ dom
𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))) |
83 | 82 | com12 32 |
. . . 4
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 →
(𝑥 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑥) = 𝑥))) |
84 | 83 | a1i 11 |
. . 3
⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 →
(𝑥 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑥) = 𝑥)))) |
85 | 13, 20, 84 | tfis3 7603 |
. 2
⊢ (𝐴 ∈ On → (𝐴 ∈ dom
𝑅1 → (𝐴 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))) |
86 | 6, 85 | mpcom 38 |
1
⊢ (𝐴 ∈ dom
𝑅1 → (𝐴 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)) |