Step | Hyp | Ref
| Expression |
1 | | oveq2 7151 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o
∅)) |
2 | 1 | eleq2d 2836 |
. . . . 5
⊢ (𝑥 = ∅ → (∅
∈ (𝐴
↑o 𝑥)
↔ ∅ ∈ (𝐴
↑o ∅))) |
3 | | oveq2 7151 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦)) |
4 | 3 | eleq2d 2836 |
. . . . 5
⊢ (𝑥 = 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝑦))) |
5 | | oveq2 7151 |
. . . . . 6
⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦)) |
6 | 5 | eleq2d 2836 |
. . . . 5
⊢ (𝑥 = suc 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o suc 𝑦))) |
7 | | oveq2 7151 |
. . . . . 6
⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵)) |
8 | 7 | eleq2d 2836 |
. . . . 5
⊢ (𝑥 = 𝐵 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝐵))) |
9 | | 0lt1o 8132 |
. . . . . . 7
⊢ ∅
∈ 1o |
10 | | oe0 8150 |
. . . . . . 7
⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) =
1o) |
11 | 9, 10 | eleqtrrid 2858 |
. . . . . 6
⊢ (𝐴 ∈ On → ∅ ∈
(𝐴 ↑o
∅)) |
12 | 11 | adantr 485 |
. . . . 5
⊢ ((𝐴 ∈ On ∧ ∅ ∈
𝐴) → ∅ ∈
(𝐴 ↑o
∅)) |
13 | | oecl 8165 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On) |
14 | | omordi 8195 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅
∈ (𝐴
↑o 𝑦))
→ (∅ ∈ 𝐴
→ ((𝐴
↑o 𝑦)
·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴))) |
15 | | om0 8145 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o ∅)
= ∅) |
16 | 15 | eleq1d 2835 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ↑o 𝑦) ∈ On → (((𝐴 ↑o 𝑦) ·o ∅)
∈ ((𝐴
↑o 𝑦)
·o 𝐴)
↔ ∅ ∈ ((𝐴
↑o 𝑦)
·o 𝐴))) |
17 | 16 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅
∈ (𝐴
↑o 𝑦))
→ (((𝐴
↑o 𝑦)
·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴))) |
18 | 14, 17 | sylibd 242 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅
∈ (𝐴
↑o 𝑦))
→ (∅ ∈ 𝐴
→ ∅ ∈ ((𝐴
↑o 𝑦)
·o 𝐴))) |
19 | 13, 18 | syldanl 605 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈
(𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴))) |
20 | | oesuc 8155 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴)) |
21 | 20 | eleq2d 2836 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅
∈ (𝐴
↑o suc 𝑦)
↔ ∅ ∈ ((𝐴
↑o 𝑦)
·o 𝐴))) |
22 | 21 | adantr 485 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈
(𝐴 ↑o 𝑦)) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴))) |
23 | 19, 22 | sylibrd 262 |
. . . . . . . . 9
⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈
(𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o suc 𝑦))) |
24 | 23 | exp31 424 |
. . . . . . . 8
⊢ (𝐴 ∈ On → (𝑦 ∈ On → (∅
∈ (𝐴
↑o 𝑦)
→ (∅ ∈ 𝐴
→ ∅ ∈ (𝐴
↑o suc 𝑦))))) |
25 | 24 | com12 32 |
. . . . . . 7
⊢ (𝑦 ∈ On → (𝐴 ∈ On → (∅
∈ (𝐴
↑o 𝑦)
→ (∅ ∈ 𝐴
→ ∅ ∈ (𝐴
↑o suc 𝑦))))) |
26 | 25 | com34 91 |
. . . . . 6
⊢ (𝑦 ∈ On → (𝐴 ∈ On → (∅
∈ 𝐴 → (∅
∈ (𝐴
↑o 𝑦)
→ ∅ ∈ (𝐴
↑o suc 𝑦))))) |
27 | 26 | impd 415 |
. . . . 5
⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈
𝐴) → (∅ ∈
(𝐴 ↑o 𝑦) → ∅ ∈ (𝐴 ↑o suc 𝑦)))) |
28 | | 0ellim 6224 |
. . . . . . . . . . . 12
⊢ (Lim
𝑥 → ∅ ∈
𝑥) |
29 | | eqimss2 3945 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ↑o ∅) =
1o → 1o ⊆ (𝐴 ↑o
∅)) |
30 | 10, 29 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ On → 1o
⊆ (𝐴
↑o ∅)) |
31 | | oveq2 7151 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ∅ → (𝐴 ↑o 𝑦) = (𝐴 ↑o
∅)) |
32 | 31 | sseq2d 3920 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ∅ → (1o
⊆ (𝐴
↑o 𝑦)
↔ 1o ⊆ (𝐴 ↑o
∅))) |
33 | 32 | rspcev 3539 |
. . . . . . . . . . . 12
⊢ ((∅
∈ 𝑥 ∧
1o ⊆ (𝐴
↑o ∅)) → ∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦)) |
34 | 28, 30, 33 | syl2an 599 |
. . . . . . . . . . 11
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → ∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦)) |
35 | | ssiun 4928 |
. . . . . . . . . . 11
⊢
(∃𝑦 ∈
𝑥 1o ⊆
(𝐴 ↑o 𝑦) → 1o ⊆
∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)) |
36 | 34, 35 | syl 17 |
. . . . . . . . . 10
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → 1o ⊆
∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)) |
37 | 36 | adantrr 717 |
. . . . . . . . 9
⊢ ((Lim
𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆
∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)) |
38 | | vex 3411 |
. . . . . . . . . . . 12
⊢ 𝑥 ∈ V |
39 | | oelim 8162 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)) |
40 | 38, 39 | mpanlr1 706 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)) |
41 | 40 | anasss 471 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)) |
42 | 41 | an12s 649 |
. . . . . . . . 9
⊢ ((Lim
𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)) |
43 | 37, 42 | sseqtrrd 3929 |
. . . . . . . 8
⊢ ((Lim
𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆
(𝐴 ↑o 𝑥)) |
44 | | limelon 6225 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On) |
45 | 38, 44 | mpan 690 |
. . . . . . . . . . 11
⊢ (Lim
𝑥 → 𝑥 ∈ On) |
46 | | oecl 8165 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑o 𝑥) ∈ On) |
47 | 46 | ancoms 463 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑o 𝑥) ∈ On) |
48 | 45, 47 | sylan 584 |
. . . . . . . . . 10
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (𝐴 ↑o 𝑥) ∈ On) |
49 | | eloni 6172 |
. . . . . . . . . 10
⊢ ((𝐴 ↑o 𝑥) ∈ On → Ord (𝐴 ↑o 𝑥)) |
50 | | ordgt0ge1 8125 |
. . . . . . . . . 10
⊢ (Ord
(𝐴 ↑o 𝑥) → (∅ ∈ (𝐴 ↑o 𝑥) ↔ 1o ⊆
(𝐴 ↑o 𝑥))) |
51 | 48, 49, 50 | 3syl 18 |
. . . . . . . . 9
⊢ ((Lim
𝑥 ∧ 𝐴 ∈ On) → (∅ ∈ (𝐴 ↑o 𝑥) ↔ 1o ⊆
(𝐴 ↑o 𝑥))) |
52 | 51 | adantrr 717 |
. . . . . . . 8
⊢ ((Lim
𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴 ↑o 𝑥) ↔ 1o ⊆
(𝐴 ↑o 𝑥))) |
53 | 43, 52 | mpbird 260 |
. . . . . . 7
⊢ ((Lim
𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴 ↑o 𝑥)) |
54 | 53 | ex 417 |
. . . . . 6
⊢ (Lim
𝑥 → ((𝐴 ∈ On ∧ ∅ ∈
𝐴) → ∅ ∈
(𝐴 ↑o 𝑥))) |
55 | 54 | a1dd 50 |
. . . . 5
⊢ (Lim
𝑥 → ((𝐴 ∈ On ∧ ∅ ∈
𝐴) → (∀𝑦 ∈ 𝑥 ∅ ∈ (𝐴 ↑o 𝑦) → ∅ ∈ (𝐴 ↑o 𝑥)))) |
56 | 2, 4, 6, 8, 12, 27, 55 | tfinds3 7571 |
. . . 4
⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈
𝐴) → ∅ ∈
(𝐴 ↑o 𝐵))) |
57 | 56 | expd 420 |
. . 3
⊢ (𝐵 ∈ On → (𝐴 ∈ On → (∅
∈ 𝐴 → ∅
∈ (𝐴
↑o 𝐵)))) |
58 | 57 | com12 32 |
. 2
⊢ (𝐴 ∈ On → (𝐵 ∈ On → (∅
∈ 𝐴 → ∅
∈ (𝐴
↑o 𝐵)))) |
59 | 58 | imp31 422 |
1
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈
𝐴) → ∅ ∈
(𝐴 ↑o 𝐵)) |