| Step | Hyp | Ref
| Expression |
| 1 | | difss 4136 |
. . . 4
⊢ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 |
| 2 | | ackbij.f |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
| 3 | 2 | ackbij1lem11 10269 |
. . . 4
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ (𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 ∖ ∩
(ω ∖ 𝐴)) ∈
(𝒫 ω ∩ Fin)) |
| 4 | 1, 3 | mpan2 691 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐴 ∖
∩ (ω ∖ 𝐴)) ∈ (𝒫 ω ∩
Fin)) |
| 5 | | difss 4136 |
. . . . . . 7
⊢ (ω
∖ 𝐴) ⊆
ω |
| 6 | | omsson 7891 |
. . . . . . 7
⊢ ω
⊆ On |
| 7 | 5, 6 | sstri 3993 |
. . . . . 6
⊢ (ω
∖ 𝐴) ⊆
On |
| 8 | | ominf 9294 |
. . . . . . . 8
⊢ ¬
ω ∈ Fin |
| 9 | | elinel2 4202 |
. . . . . . . 8
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → 𝐴 ∈
Fin) |
| 10 | | difinf 9349 |
. . . . . . . 8
⊢ ((¬
ω ∈ Fin ∧ 𝐴
∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin) |
| 11 | 8, 9, 10 | sylancr 587 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ¬ (ω ∖ 𝐴) ∈ Fin) |
| 12 | | 0fi 9082 |
. . . . . . . . 9
⊢ ∅
∈ Fin |
| 13 | | eleq1 2829 |
. . . . . . . . 9
⊢ ((ω
∖ 𝐴) = ∅ →
((ω ∖ 𝐴) ∈
Fin ↔ ∅ ∈ Fin)) |
| 14 | 12, 13 | mpbiri 258 |
. . . . . . . 8
⊢ ((ω
∖ 𝐴) = ∅ →
(ω ∖ 𝐴) ∈
Fin) |
| 15 | 14 | necon3bi 2967 |
. . . . . . 7
⊢ (¬
(ω ∖ 𝐴) ∈
Fin → (ω ∖ 𝐴) ≠ ∅) |
| 16 | 11, 15 | syl 17 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (ω ∖ 𝐴) ≠ ∅) |
| 17 | | onint 7810 |
. . . . . 6
⊢
(((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴)) |
| 18 | 7, 16, 17 | sylancr 587 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ∈ (ω ∖ 𝐴)) |
| 19 | 18 | eldifad 3963 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ∈ ω) |
| 20 | | ackbij1lem4 10262 |
. . . 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 10264 |
. . 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 584 |
. 2
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∪ {∩
(ω ∖ 𝐴)})
∈ (𝒫 ω ∩ Fin)) |
| 24 | 18 | eldifbd 3964 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴) |
| 25 | | disjsn 4711 |
. . . . . 6
⊢ ((𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅ ↔ ¬ ∩ (ω ∖ 𝐴) ∈ 𝐴) |
| 26 | 24, 25 | sylibr 234 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐴 ∩ {∩ (ω ∖ 𝐴)}) = ∅) |
| 27 | | ssdisj 4460 |
. . . . 5
⊢ (((𝐴 ∖ ∩ (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ {∩
(ω ∖ 𝐴)}) =
∅) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ {∩
(ω ∖ 𝐴)}) =
∅) |
| 28 | 1, 26, 27 | sylancr 587 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ {∩
(ω ∖ 𝐴)}) =
∅) |
| 29 | 2 | ackbij1lem9 10267 |
. . . 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 1373 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘{∩
(ω ∖ 𝐴)}))) |
| 31 | 2 | ackbij1lem14 10272 |
. . . . 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 7447 |
. . 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 10268 |
. . . . . . 7
⊢ 𝐹:(𝒫 ω ∩
Fin)⟶ω |
| 35 | 34 | ffvelcdmi 7103 |
. . . . . 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 10261 |
. . . . . . 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 | ffvelcdmi 7103 |
. . . . . 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 8644 |
. . . . 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 584 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o suc (𝐹‘∩ (ω
∖ 𝐴))) = suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴)))) |
| 43 | | disjdifr 4473 |
. . . . . . . 8
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
∅ |
| 44 | 43 | a1i 11 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∩ ∩
(ω ∖ 𝐴)) =
∅) |
| 45 | 2 | ackbij1lem9 10267 |
. . . . . . 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)))) |
| 46 | 4, 38, 44, 45 | syl3anc 1373 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
∩ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴)))) |
| 47 | | uncom 4158 |
. . . . . . . 8
⊢ ((𝐴 ∖ ∩ (ω ∖ 𝐴)) ∪ ∩
(ω ∖ 𝐴)) =
(∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩
(ω ∖ 𝐴))) |
| 48 | | onnmin 7818 |
. . . . . . . . . . . . . . 15
⊢
(((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ ∩ (ω
∖ 𝐴)) |
| 49 | 7, 48 | mpan 690 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 ∈ ∩ (ω ∖ 𝐴)) |
| 50 | 49 | con2i 139 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ ∩ (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴)) |
| 51 | 50 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴)) |
| 52 | | ordom 7897 |
. . . . . . . . . . . . . . 15
⊢ Ord
ω |
| 53 | | ordelss 6400 |
. . . . . . . . . . . . . . 15
⊢ ((Ord
ω ∧ ∩ (ω ∖ 𝐴) ∈ ω) → ∩ (ω ∖ 𝐴) ⊆ ω) |
| 54 | 52, 19, 53 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ⊆ ω) |
| 55 | 54 | sselda 3983 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ ω) |
| 56 | | eldif 3961 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ 𝐴)) |
| 57 | 56 | simplbi2 500 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ ω → (¬
𝑎 ∈ 𝐴 → 𝑎 ∈ (ω ∖ 𝐴))) |
| 58 | 57 | orrd 864 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ ω → (𝑎 ∈ 𝐴 ∨ 𝑎 ∈ (ω ∖ 𝐴))) |
| 59 | 58 | orcomd 872 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ ω → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴)) |
| 60 | 55, 59 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴)) |
| 61 | | orel1 889 |
. . . . . . . . . . . 12
⊢ (¬
𝑎 ∈ (ω ∖
𝐴) → ((𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)) |
| 62 | 51, 60, 61 | sylc 65 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ (𝒫 ω ∩
Fin) ∧ 𝑎 ∈ ∩ (ω ∖ 𝐴)) → 𝑎 ∈ 𝐴) |
| 63 | 62 | ex 412 |
. . . . . . . . . 10
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝑎 ∈ ∩ (ω ∖ 𝐴) → 𝑎 ∈ 𝐴)) |
| 64 | 63 | ssrdv 3989 |
. . . . . . . . 9
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∩ (ω ∖ 𝐴) ⊆ 𝐴) |
| 65 | | undif 4482 |
. . . . . . . . 9
⊢ (∩ (ω ∖ 𝐴) ⊆ 𝐴 ↔ (∩
(ω ∖ 𝐴) ∪
(𝐴 ∖ ∩ (ω ∖ 𝐴))) = 𝐴) |
| 66 | 64, 65 | sylib 218 |
. . . . . . . 8
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (∩ (ω ∖ 𝐴) ∪ (𝐴 ∖ ∩
(ω ∖ 𝐴))) =
𝐴) |
| 67 | 47, 66 | eqtrid 2789 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐴 ∖
∩ (ω ∖ 𝐴)) ∪ ∩
(ω ∖ 𝐴)) =
𝐴) |
| 68 | 67 | fveq2d 6910 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
∩ (ω ∖ 𝐴))) = (𝐹‘𝐴)) |
| 69 | 46, 68 | eqtr3d 2779 |
. . . . 5
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴))) = (𝐹‘𝐴)) |
| 70 | | suceq 6450 |
. . . . 5
⊢ (((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴))) = (𝐹‘𝐴) → suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
| 71 | 69, 70 | syl 17 |
. . . 4
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → suc ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
| 72 | 42, 71 | eqtrd 2777 |
. . 3
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ((𝐹‘(𝐴 ∖ ∩
(ω ∖ 𝐴)))
+o suc (𝐹‘∩ (ω
∖ 𝐴))) = suc (𝐹‘𝐴)) |
| 73 | 30, 33, 72 | 3eqtrd 2781 |
. 2
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴)) |
| 74 | | fveqeq2 6915 |
. . 3
⊢ (𝑏 = ((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)}) → ((𝐹‘𝑏) = suc (𝐹‘𝐴) ↔ (𝐹‘((𝐴 ∖ ∩
(ω ∖ 𝐴)) ∪
{∩ (ω ∖ 𝐴)})) = suc (𝐹‘𝐴))) |
| 75 | 74 | rspcev 3622 |
. 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)) |
| 76 | 23, 73, 75 | syl2anc 584 |
1
⊢ (𝐴 ∈ (𝒫 ω ∩
Fin) → ∃𝑏 ∈
(𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴)) |