Step | Hyp | Ref
| Expression |
1 | | difss 3737 |
. . . 4
⊢ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 |
2 | | ackbij.f |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
3 | 2 | ackbij1lem11 9052 |
. . . 4
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 ∖ ∩
(ω ∖ 𝐴)) ∈
(𝒫 ω ∩ Fin)) |
4 | 1, 3 | mpan2 707 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐴 ∖
∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩
Fin)) |
5 | | difss 3737 |
. . . . . . 7
⊢ (ω
∖ 𝐴) ⊆
ω |
6 | | omsson 7069 |
. . . . . . 7
⊢ ω
⊆ On |
7 | 5, 6 | sstri 3612 |
. . . . . 6
⊢ (ω
∖ 𝐴) ⊆
On |
8 | | ominf 8172 |
. . . . . . . 8
⊢ ¬
ω ∈ Fin |
9 | | inss2 3834 |
. . . . . . . . 9
⊢
(𝒫 ω ∩ Fin) ⊆ Fin |
10 | 9 | sseli 3599 |
. . . . . . . 8
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → 𝐴 ∈
Fin) |
11 | | difinf 8230 |
. . . . . . . 8
⊢ ((¬
ω ∈ Fin ∧ 𝐴
∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin) |
12 | 8, 10, 11 | sylancr 695 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ¬ (ω ∖ 𝐴) ∈ Fin) |
13 | | 0fin 8188 |
. . . . . . . . 9
⊢ ∅
∈ Fin |
14 | | eleq1 2689 |
. . . . . . . . 9
⊢ ((ω
∖ 𝐴) = ∅ →
((ω ∖ 𝐴) ∈
Fin ↔ ∅ ∈ Fin)) |
15 | 13, 14 | mpbiri 248 |
. . . . . . . 8
⊢ ((ω
∖ 𝐴) = ∅ →
(ω ∖ 𝐴) ∈
Fin) |
16 | 15 | necon3bi 2820 |
. . . . . . 7
⊢ (¬
(ω ∖ 𝐴) ∈
Fin → (ω ∖ 𝐴) ≠ ∅) |
17 | 12, 16 | syl 17 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (ω ∖ 𝐴) ≠ ∅) |
18 | | onint 6995 |
. . . . . 6
⊢
(((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴)) |
19 | 7, 17, 18 | sylancr 695 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴)) |
20 | 19 | eldifad 3586 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ∈ ω) |
21 | | ackbij1lem4 9045 |
. . . 4
⊢ (∩ (ω ∖ 𝐴) ∈ ω → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩
Fin)) |
22 | 20, 21 | syl 17 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩
Fin)) |
23 | | ackbij1lem6 9047 |
. . 3
⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin)
∧ {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
→ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩
(ω ∖ 𝐴)})
∈ (𝒫 ω ∩ Fin)) |
24 | 4, 22, 23 | syl2anc 693 |
. 2
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∪ {∩
(ω ∖ 𝐴)})
∈ (𝒫 ω ∩ Fin)) |
25 | 19 | eldifbd 3587 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴) |
26 | | disjsn 4246 |
. . . . . 6
⊢ ((𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅ ↔ ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴) |
27 | 25, 26 | sylibr 224 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅) |
28 | | ssdisj 4026 |
. . . . 5
⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ {∩
(ω ∖ 𝐴)}) =
∅) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ {∩
(ω ∖ 𝐴)}) =
∅) |
29 | 1, 27, 28 | sylancr 695 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ {∩
(ω ∖ 𝐴)}) =
∅) |
30 | 2 | ackbij1lem9 9050 |
. . . 4
⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin)
∧ {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin)
∧ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ {∩
(ω ∖ 𝐴)}) =
∅) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘{∩
(ω ∖ 𝐴)}))) |
31 | 4, 22, 29, 30 | syl3anc 1326 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘{∩
(ω ∖ 𝐴)}))) |
32 | 2 | ackbij1lem14 9055 |
. . . . 5
⊢ (∩ (ω ∖ 𝐴) ∈ ω → (𝐹‘{∩
(ω ∖ 𝐴)}) = suc
(𝐹‘∩ (ω ∖ 𝐴))) |
33 | 20, 32 | syl 17 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘{∩ (ω ∖ 𝐴)}) = suc (𝐹‘∩ (ω
∖ 𝐴))) |
34 | 33 | oveq2d 6666 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘{∩
(ω ∖ 𝐴)})) =
((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 suc (𝐹‘∩ (ω ∖ 𝐴)))) |
35 | 2 | ackbij1lem10 9051 |
. . . . . . 7
⊢ 𝐹:(𝒫 ω ∩
Fin)⟶ω |
36 | 35 | ffvelrni 6358 |
. . . . . 6
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin)
→ (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω) |
37 | 4, 36 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω) |
38 | | ackbij1lem3 9044 |
. . . . . . 7
⊢ (∩ (ω ∖ 𝐴) ∈ ω → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩
Fin)) |
39 | 20, 38 | syl 17 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩
Fin)) |
40 | 35 | ffvelrni 6358 |
. . . . . 6
⊢ (∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin)
→ (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω) |
41 | 39, 40 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω) |
42 | | nnasuc 7686 |
. . . . 5
⊢ (((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
∈ ω ∧ (𝐹‘∩ (ω
∖ 𝐴)) ∈ ω)
→ ((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +𝑜 suc (𝐹‘∩ (ω ∖ 𝐴))) = suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘∩ (ω
∖ 𝐴)))) |
43 | 37, 41, 42 | syl2anc 693 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 suc (𝐹‘∩ (ω
∖ 𝐴))) = suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘∩ (ω
∖ 𝐴)))) |
44 | | incom 3805 |
. . . . . . . . 9
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
(∩ (ω ∖ 𝐴) ∩ (𝐴 ∖ ∩
(ω ∖ 𝐴))) |
45 | | disjdif 4040 |
. . . . . . . . 9
⊢ (∩ (ω ∖ 𝐴) ∩ (𝐴 ∖ ∩
(ω ∖ 𝐴))) =
∅ |
46 | 44, 45 | eqtri 2644 |
. . . . . . . 8
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
∅ |
47 | 46 | a1i 11 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
∅) |
48 | 2 | ackbij1lem9 9050 |
. . . . . . 7
⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin)
∧ ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) ∧
((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
∅) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘∩ (ω
∖ 𝐴)))) |
49 | 4, 39, 47, 48 | syl3anc 1326 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘∩ (ω
∖ 𝐴)))) |
50 | | uncom 3757 |
. . . . . . . 8
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩
(ω ∖ 𝐴)) =
(∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩
(ω ∖ 𝐴))) |
51 | | onnmin 7003 |
. . . . . . . . . . . . . . 15
⊢
(((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ ∩ (ω
∖ 𝐴)) |
52 | 7, 51 | mpan 706 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴)) |
53 | 52 | con2i 134 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ ∩ (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴)) |
54 | 53 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴)) |
55 | | ordom 7074 |
. . . . . . . . . . . . . . 15
⊢ Ord
ω |
56 | | ordelss 5739 |
. . . . . . . . . . . . . . 15
⊢ ((Ord
ω ∧ ∩ (ω ∖ 𝐴) ∈ ω) → ∩ (ω ∖ 𝐴) ⊆ ω) |
57 | 55, 20, 56 | sylancr 695 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ⊆ ω) |
58 | 57 | sselda 3603 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ ω) |
59 | | eldif 3584 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ 𝐴)) |
60 | 59 | simplbi2 655 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ ω → (¬
𝑎 ∈ 𝐴 → 𝑎 ∈ (ω ∖ 𝐴))) |
61 | 60 | orrd 393 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ ω → (𝑎 ∈ 𝐴 ∨ 𝑎 ∈ (ω ∖ 𝐴))) |
62 | 61 | orcomd 403 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ ω → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴)) |
63 | 58, 62 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴)) |
64 | | orel1 397 |
. . . . . . . . . . . 12
⊢ (¬
𝑎 ∈ (ω ∖
𝐴) → ((𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)) |
65 | 54, 63, 64 | sylc 65 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ 𝐴) |
66 | 65 | ex 450 |
. . . . . . . . . 10
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝑎 ∈ ∩ (ω ∖ 𝐴) → 𝑎 ∈ 𝐴)) |
67 | 66 | ssrdv 3609 |
. . . . . . . . 9
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ⊆ 𝐴) |
68 | | undif 4049 |
. . . . . . . . 9
⊢ (∩ (ω ∖ 𝐴) ⊆ 𝐴 ↔ (∩
(ω ∖ 𝐴) ∪
(𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴) |
69 | 67, 68 | sylib 208 |
. . . . . . . 8
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩
(ω ∖ 𝐴))) =
𝐴) |
70 | 50, 69 | syl5eq 2668 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∪ ∩
(ω ∖ 𝐴)) =
𝐴) |
71 | 70 | fveq2d 6195 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
∩ (ω ∖ 𝐴))) = (𝐹‘𝐴)) |
72 | 49, 71 | eqtr3d 2658 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘∩ (ω
∖ 𝐴))) = (𝐹‘𝐴)) |
73 | | suceq 5790 |
. . . . 5
⊢ (((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘∩ (ω
∖ 𝐴))) = (𝐹‘𝐴) → suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
74 | 72, 73 | syl 17 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
75 | 43, 74 | eqtrd 2656 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+𝑜 suc (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
76 | 31, 34, 75 | 3eqtrd 2660 |
. 2
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)) |
77 | | fveq2 6191 |
. . . 4
⊢ (𝑏 = ((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)}) → (𝐹‘𝑏) = (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)}))) |
78 | 77 | eqeq1d 2624 |
. . 3
⊢ (𝑏 = ((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)}) → ((𝐹‘𝑏) = suc (𝐹‘𝐴) ↔ (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴))) |
79 | 78 | rspcev 3309 |
. 2
⊢ ((((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩
(ω ∖ 𝐴)})
∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴)) |
80 | 24, 76, 79 | syl2anc 693 |
1
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∃𝑏 ∈
(𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴)) |