Step | Hyp | Ref
| Expression |
1 | | simp2 1062 |
. . 3
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ∈ On) |
2 | | sucelon 7017 |
. . . . . 6
⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On) |
3 | 1, 2 | sylib 208 |
. . . . 5
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → suc 𝐵 ∈ On) |
4 | | simp1 1061 |
. . . . 5
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On) |
5 | | on0eln0 5780 |
. . . . . . 7
⊢ (𝐴 ∈ On → (∅
∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
6 | 5 | biimpar 502 |
. . . . . 6
⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅) → ∅
∈ 𝐴) |
7 | 6 | 3adant2 1080 |
. . . . 5
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∅
∈ 𝐴) |
8 | | omword2 7654 |
. . . . 5
⊢ (((suc
𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈
𝐴) → suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵)) |
9 | 3, 4, 7, 8 | syl21anc 1325 |
. . . 4
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵)) |
10 | | sucidg 5803 |
. . . . 5
⊢ (𝐵 ∈ On → 𝐵 ∈ suc 𝐵) |
11 | | ssel 3597 |
. . . . 5
⊢ (suc
𝐵 ⊆ (𝐴 ·𝑜
suc 𝐵) → (𝐵 ∈ suc 𝐵 → 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵))) |
12 | 10, 11 | syl5 34 |
. . . 4
⊢ (suc
𝐵 ⊆ (𝐴 ·𝑜
suc 𝐵) → (𝐵 ∈ On → 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵))) |
13 | 9, 1, 12 | sylc 65 |
. . 3
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)) |
14 | | suceq 5790 |
. . . . . 6
⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) |
15 | 14 | oveq2d 6666 |
. . . . 5
⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 suc 𝑥) = (𝐴 ·𝑜 suc 𝐵)) |
16 | 15 | eleq2d 2687 |
. . . 4
⊢ (𝑥 = 𝐵 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵))) |
17 | 16 | rspcev 3309 |
. . 3
⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥)) |
18 | 1, 13, 17 | syl2anc 693 |
. 2
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) →
∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥)) |
19 | | suceq 5790 |
. . . . . 6
⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧) |
20 | 19 | oveq2d 6666 |
. . . . 5
⊢ (𝑥 = 𝑧 → (𝐴 ·𝑜 suc 𝑥) = (𝐴 ·𝑜 suc 𝑧)) |
21 | 20 | eleq2d 2687 |
. . . 4
⊢ (𝑥 = 𝑧 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))) |
22 | 21 | onminex 7007 |
. . 3
⊢
(∃𝑥 ∈ On
𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) → ∃𝑥 ∈ On (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))) |
23 | | vex 3203 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ∈ V |
24 | 23 | elon 5732 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
25 | | ordzsl 7045 |
. . . . . . . . . . . . . 14
⊢ (Ord
𝑥 ↔ (𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥)) |
26 | 24, 25 | bitri 264 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥)) |
27 | | noel 3919 |
. . . . . . . . . . . . . . . 16
⊢ ¬
𝐵 ∈
∅ |
28 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = ∅ → (𝐴 ·𝑜
𝑥) = (𝐴 ·𝑜
∅)) |
29 | | om0x 7599 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ·𝑜
∅) = ∅ |
30 | 28, 29 | syl6eq 2672 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = ∅ → (𝐴 ·𝑜
𝑥) =
∅) |
31 | 30 | eleq2d 2687 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ∅ → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ ∅)) |
32 | 27, 31 | mtbiri 317 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = ∅ → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)) |
33 | 32 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 = ∅ → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
34 | | simp3 1063 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → 𝑥 = suc 𝑤) |
35 | | simp2 1062 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) |
36 | | raleq 3138 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = suc 𝑤 → (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))) |
37 | | vex 3203 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑤 ∈ V |
38 | 37 | sucid 5804 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑤 ∈ suc 𝑤 |
39 | | suceq 5790 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 = 𝑤 → suc 𝑧 = suc 𝑤) |
40 | 39 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 = 𝑤 → (𝐴 ·𝑜 suc 𝑧) = (𝐴 ·𝑜 suc 𝑤)) |
41 | 40 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = 𝑤 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))) |
42 | 41 | notbid 308 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑤 → (¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))) |
43 | 42 | rspcv 3305 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∈ suc 𝑤 → (∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))) |
44 | 38, 43 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑧 ∈
suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)) |
45 | 36, 44 | syl6bi 243 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = suc 𝑤 → (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))) |
46 | 34, 35, 45 | sylc 65 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)) |
47 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = suc 𝑤 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑤)) |
48 | 47 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = suc 𝑤 → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))) |
49 | 48 | notbid 308 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = suc 𝑤 → (¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))) |
50 | 49 | biimpar 502 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 = suc 𝑤 ∧ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)) |
51 | 34, 46, 50 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)) |
52 | 51 | 3expia 1267 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 = suc 𝑤 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
53 | 52 | rexlimdvw 3034 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (∃𝑤 ∈ On 𝑥 = suc 𝑤 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
54 | | ralnex 2992 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑧 ∈
𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ¬ ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) |
55 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → 𝐴 ∈ On) |
56 | 23 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → 𝑥 ∈ V) |
57 | | simpl 473 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → Lim 𝑥) |
58 | | omlim 7613 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧)) |
59 | 55, 56, 57, 58 | syl12anc 1324 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝑥) = ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧)) |
60 | 59 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ ∪
𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧))) |
61 | | eliun 4524 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐵 ∈ ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ↔ ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 𝑧)) |
62 | | limord 5784 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (Lim
𝑥 → Ord 𝑥) |
63 | 62 | 3ad2ant1 1082 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → Ord 𝑥) |
64 | 63, 24 | sylibr 224 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ On) |
65 | | simp3 1063 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥) |
66 | | onelon 5748 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On) |
67 | 64, 65, 66 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On) |
68 | | suceloni 7013 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 ∈ On → suc 𝑧 ∈ On) |
69 | 67, 68 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → suc 𝑧 ∈ On) |
70 | | simp2 1062 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝐴 ∈ On) |
71 | | sssucid 5802 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 𝑧 ⊆ suc 𝑧 |
72 | | omwordi 7651 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑧 ∈ On ∧ suc 𝑧 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ suc 𝑧 → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 suc 𝑧))) |
73 | 71, 72 | mpi 20 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑧 ∈ On ∧ suc 𝑧 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜
𝑧) ⊆ (𝐴 ·𝑜
suc 𝑧)) |
74 | 67, 69, 70, 73 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 suc 𝑧)) |
75 | 74 | sseld 3602 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → (𝐵 ∈ (𝐴 ·𝑜 𝑧) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))) |
76 | 75 | 3expia 1267 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (𝑧 ∈ 𝑥 → (𝐵 ∈ (𝐴 ·𝑜 𝑧) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))) |
77 | 76 | reximdvai 3015 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 𝑧) → ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))) |
78 | 61, 77 | syl5bi 232 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (𝐵 ∈ ∪
𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧) → ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))) |
79 | 60, 78 | sylbid 230 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (𝐵 ∈ (𝐴 ·𝑜 𝑥) → ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))) |
80 | 79 | con3d 148 |
. . . . . . . . . . . . . . . . . 18
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (¬ ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
81 | 54, 80 | syl5bi 232 |
. . . . . . . . . . . . . . . . 17
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
82 | 81 | expimpd 629 |
. . . . . . . . . . . . . . . 16
⊢ (Lim
𝑥 → ((𝐴 ∈ On ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
83 | 82 | com12 32 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ On ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (Lim 𝑥 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
84 | 83 | 3ad2antl1 1223 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (Lim 𝑥 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
85 | 33, 53, 84 | 3jaod 1392 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ((𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
86 | 26, 85 | syl5bi 232 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 ∈ On → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
87 | 86 | impr 649 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)) |
88 | | simpl1 1064 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝐴 ∈ On) |
89 | | simprr 796 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝑥 ∈ On) |
90 | | omcl 7616 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜
𝑥) ∈
On) |
91 | 88, 89, 90 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (𝐴 ·𝑜 𝑥) ∈ On) |
92 | | simpl2 1065 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝐵 ∈ On) |
93 | | ontri1 5757 |
. . . . . . . . . . . 12
⊢ (((𝐴 ·𝑜
𝑥) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜
𝑥) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
94 | 91, 92, 93 | syl2anc 693 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))) |
95 | 87, 94 | mpbird 247 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (𝐴 ·𝑜 𝑥) ⊆ 𝐵) |
96 | | oawordex 7637 |
. . . . . . . . . . 11
⊢ (((𝐴 ·𝑜
𝑥) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜
𝑥) ⊆ 𝐵 ↔ ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) |
97 | 91, 92, 96 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) |
98 | 95, 97 | mpbid 222 |
. . . . . . . . 9
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧
(∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵) |
99 | 98 | 3adantr1 1220 |
. . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵) |
100 | | simp3r 1090 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵) |
101 | | simp21 1094 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥)) |
102 | | simp11 1091 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝐴 ∈ On) |
103 | | simp23 1096 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝑥 ∈ On) |
104 | | omsuc 7606 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜
suc 𝑥) = ((𝐴 ·𝑜
𝑥) +𝑜
𝐴)) |
105 | 102, 103,
104 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) |
106 | 101, 105 | eleqtrd 2703 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝐵 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) |
107 | 100, 106 | eqeltrd 2701 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) |
108 | | simp3l 1089 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝑦 ∈ On) |
109 | 102, 103,
90 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝐴 ·𝑜 𝑥) ∈ On) |
110 | | oaord 7627 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜
𝑥) ∈ On) → (𝑦 ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))) |
111 | 108, 102,
109, 110 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝑦 ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))) |
112 | 107, 111 | mpbird 247 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝑦 ∈ 𝐴) |
113 | 112, 100 | jca 554 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝑦 ∈ 𝐴 ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) |
114 | 113 | 3expia 1267 |
. . . . . . . . 9
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵) → (𝑦 ∈ 𝐴 ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))) |
115 | 114 | reximdv2 3014 |
. . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵 → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) |
116 | 99, 115 | mpd 15 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵) |
117 | 116 | expcom 451 |
. . . . . 6
⊢ ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) |
118 | 117 | 3expia 1267 |
. . . . 5
⊢ ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))) |
119 | 118 | com13 88 |
. . . 4
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (𝑥 ∈ On → ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))) |
120 | 119 | reximdvai 3015 |
. . 3
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) →
(∃𝑥 ∈ On (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) |
121 | 22, 120 | syl5 34 |
. 2
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) →
(∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) |
122 | 18, 121 | mpd 15 |
1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) →
∃𝑥 ∈ On
∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵) |