Step | Hyp | Ref
| Expression |
1 | | difss 4110 |
. . . 4
⊢ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 |
2 | | ackbij.f |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
3 | 2 | ackbij1lem11 9654 |
. . . 4
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 ∖ ∩
(ω ∖ 𝐴)) ∈
(𝒫 ω ∩ Fin)) |
4 | 1, 3 | mpan2 689 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐴 ∖
∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩
Fin)) |
5 | | difss 4110 |
. . . . . . 7
⊢ (ω
∖ 𝐴) ⊆
ω |
6 | | omsson 7586 |
. . . . . . 7
⊢ ω
⊆ On |
7 | 5, 6 | sstri 3978 |
. . . . . 6
⊢ (ω
∖ 𝐴) ⊆
On |
8 | | ominf 8732 |
. . . . . . . 8
⊢ ¬
ω ∈ Fin |
9 | | elinel2 4175 |
. . . . . . . 8
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → 𝐴 ∈
Fin) |
10 | | difinf 8790 |
. . . . . . . 8
⊢ ((¬
ω ∈ Fin ∧ 𝐴
∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin) |
11 | 8, 9, 10 | sylancr 589 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ¬ (ω ∖ 𝐴) ∈ Fin) |
12 | | 0fin 8748 |
. . . . . . . . 9
⊢ ∅
∈ Fin |
13 | | eleq1 2902 |
. . . . . . . . 9
⊢ ((ω
∖ 𝐴) = ∅ →
((ω ∖ 𝐴) ∈
Fin ↔ ∅ ∈ Fin)) |
14 | 12, 13 | mpbiri 260 |
. . . . . . . 8
⊢ ((ω
∖ 𝐴) = ∅ →
(ω ∖ 𝐴) ∈
Fin) |
15 | 14 | necon3bi 3044 |
. . . . . . 7
⊢ (¬
(ω ∖ 𝐴) ∈
Fin → (ω ∖ 𝐴) ≠ ∅) |
16 | 11, 15 | syl 17 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (ω ∖ 𝐴) ≠ ∅) |
17 | | onint 7512 |
. . . . . 6
⊢
(((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴)) |
18 | 7, 16, 17 | sylancr 589 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴)) |
19 | 18 | eldifad 3950 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ∈ ω) |
20 | | ackbij1lem4 9647 |
. . . 4
⊢ (∩ (ω ∖ 𝐴) ∈ ω → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩
Fin)) |
21 | 19, 20 | syl 17 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩
Fin)) |
22 | | ackbij1lem6 9649 |
. . 3
⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin)
∧ {∩ (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
→ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ {∩
(ω ∖ 𝐴)})
∈ (𝒫 ω ∩ Fin)) |
23 | 4, 21, 22 | syl2anc 586 |
. 2
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∪ {∩
(ω ∖ 𝐴)})
∈ (𝒫 ω ∩ Fin)) |
24 | 18 | eldifbd 3951 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴) |
25 | | disjsn 4649 |
. . . . . 6
⊢ ((𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅ ↔ ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴) |
26 | 24, 25 | sylibr 236 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅) |
27 | | ssdisj 4411 |
. . . . 5
⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ {∩
(ω ∖ 𝐴)}) =
∅) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ {∩
(ω ∖ 𝐴)}) =
∅) |
28 | 1, 26, 27 | sylancr 589 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ {∩
(ω ∖ 𝐴)}) =
∅) |
29 | 2 | ackbij1lem9 9652 |
. . . 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)}))) |
30 | 4, 21, 28, 29 | syl3anc 1367 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘{∩
(ω ∖ 𝐴)}))) |
31 | 2 | ackbij1lem14 9657 |
. . . . 5
⊢ (∩ (ω ∖ 𝐴) ∈ ω → (𝐹‘{∩
(ω ∖ 𝐴)}) = suc
(𝐹‘∩ (ω ∖ 𝐴))) |
32 | 19, 31 | syl 17 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘{∩ (ω ∖ 𝐴)}) = suc (𝐹‘∩ (ω
∖ 𝐴))) |
33 | 32 | oveq2d 7174 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘{∩
(ω ∖ 𝐴)})) =
((𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) +o suc (𝐹‘∩ (ω
∖ 𝐴)))) |
34 | 2 | ackbij1lem10 9653 |
. . . . . . 7
⊢ 𝐹:(𝒫 ω ∩
Fin)⟶ω |
35 | 34 | ffvelrni 6852 |
. . . . . 6
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin)
→ (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω) |
36 | 4, 35 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘(𝐴 ∖ ∩ (ω ∖ 𝐴))) ∈ ω) |
37 | | ackbij1lem3 9646 |
. . . . . . 7
⊢ (∩ (ω ∖ 𝐴) ∈ ω → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩
Fin)) |
38 | 19, 37 | syl 17 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩
Fin)) |
39 | 34 | ffvelrni 6852 |
. . . . . 6
⊢ (∩ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin)
→ (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω) |
40 | 38, 39 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘∩ (ω ∖ 𝐴)) ∈ ω) |
41 | | nnasuc 8234 |
. . . . 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)))) |
42 | 36, 40, 41 | syl2anc 586 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o suc (𝐹‘∩ (ω
∖ 𝐴))) = suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴)))) |
43 | | incom 4180 |
. . . . . . . . 9
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
(∩ (ω ∖ 𝐴) ∩ (𝐴 ∖ ∩
(ω ∖ 𝐴))) |
44 | | disjdif 4423 |
. . . . . . . . 9
⊢ (∩ (ω ∖ 𝐴) ∩ (𝐴 ∖ ∩
(ω ∖ 𝐴))) =
∅ |
45 | 43, 44 | eqtri 2846 |
. . . . . . . 8
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
∅ |
46 | 45 | a1i 11 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
∅) |
47 | 2 | ackbij1lem9 9652 |
. . . . . . 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)))) |
48 | 4, 38, 46, 47 | syl3anc 1367 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴)))) |
49 | | uncom 4131 |
. . . . . . . 8
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩
(ω ∖ 𝐴)) =
(∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩
(ω ∖ 𝐴))) |
50 | | onnmin 7520 |
. . . . . . . . . . . . . . 15
⊢
(((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ ∩ (ω
∖ 𝐴)) |
51 | 7, 50 | mpan 688 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴)) |
52 | 51 | con2i 141 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ ∩ (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴)) |
53 | 52 | adantl 484 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴)) |
54 | | ordom 7591 |
. . . . . . . . . . . . . . 15
⊢ Ord
ω |
55 | | ordelss 6209 |
. . . . . . . . . . . . . . 15
⊢ ((Ord
ω ∧ ∩ (ω ∖ 𝐴) ∈ ω) → ∩ (ω ∖ 𝐴) ⊆ ω) |
56 | 54, 19, 55 | sylancr 589 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ⊆ ω) |
57 | 56 | sselda 3969 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ ω) |
58 | | eldif 3948 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ 𝐴)) |
59 | 58 | simplbi2 503 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ ω → (¬
𝑎 ∈ 𝐴 → 𝑎 ∈ (ω ∖ 𝐴))) |
60 | 59 | orrd 859 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ ω → (𝑎 ∈ 𝐴 ∨ 𝑎 ∈ (ω ∖ 𝐴))) |
61 | 60 | orcomd 867 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ ω → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴)) |
62 | 57, 61 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴)) |
63 | | orel1 885 |
. . . . . . . . . . . 12
⊢ (¬
𝑎 ∈ (ω ∖
𝐴) → ((𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)) |
64 | 53, 62, 63 | sylc 65 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ 𝐴) |
65 | 64 | ex 415 |
. . . . . . . . . 10
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝑎 ∈ ∩ (ω ∖ 𝐴) → 𝑎 ∈ 𝐴)) |
66 | 65 | ssrdv 3975 |
. . . . . . . . 9
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ⊆ 𝐴) |
67 | | undif 4432 |
. . . . . . . . 9
⊢ (∩ (ω ∖ 𝐴) ⊆ 𝐴 ↔ (∩
(ω ∖ 𝐴) ∪
(𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴) |
68 | 66, 67 | sylib 220 |
. . . . . . . 8
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩
(ω ∖ 𝐴))) =
𝐴) |
69 | 49, 68 | syl5eq 2870 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∪ ∩
(ω ∖ 𝐴)) =
𝐴) |
70 | 69 | fveq2d 6676 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
∩ (ω ∖ 𝐴))) = (𝐹‘𝐴)) |
71 | 48, 70 | eqtr3d 2860 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴))) = (𝐹‘𝐴)) |
72 | | suceq 6258 |
. . . . 5
⊢ (((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴))) = (𝐹‘𝐴) → suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
73 | 71, 72 | syl 17 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
74 | 42, 73 | eqtrd 2858 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o suc (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
75 | 30, 33, 74 | 3eqtrd 2862 |
. 2
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)) |
76 | | fveqeq2 6681 |
. . 3
⊢ (𝑏 = ((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)}) → ((𝐹‘𝑏) = suc (𝐹‘𝐴) ↔ (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴))) |
77 | 76 | rspcev 3625 |
. 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)) |
78 | 23, 75, 77 | syl2anc 586 |
1
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∃𝑏 ∈
(𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴)) |