Step | Hyp | Ref
| Expression |
1 | | fin23lem33.f |
. . 3
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran
𝑎 ∈ ran 𝑎)} |
2 | | fin23lem.f |
. . 3
⊢ (𝜑 → ℎ:ω–1-1→V) |
3 | | fin23lem.g |
. . 3
⊢ (𝜑 → ∪ ran ℎ
⊆ 𝐺) |
4 | | fin23lem.h |
. . 3
⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran
𝑗))) |
5 | | fin23lem.i |
. . 3
⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω) |
6 | 1, 2, 3, 4, 5 | fin23lem38 9990 |
. 2
⊢ (𝜑 → ¬ ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
7 | 1, 2, 3, 4, 5 | fin23lem35 9988 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊊ ∪ ran
(𝑌‘𝑒)) |
8 | 7 | pssssd 4028 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊆ ∪ ran
(𝑌‘𝑒)) |
9 | | peano2 7689 |
. . . . . . . . 9
⊢ (𝑒 ∈ ω → suc 𝑒 ∈
ω) |
10 | | fveq2 6738 |
. . . . . . . . . . . 12
⊢ (𝑐 = suc 𝑒 → (𝑌‘𝑐) = (𝑌‘suc 𝑒)) |
11 | 10 | rneqd 5824 |
. . . . . . . . . . 11
⊢ (𝑐 = suc 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘suc 𝑒)) |
12 | 11 | unieqd 4849 |
. . . . . . . . . 10
⊢ (𝑐 = suc 𝑒 → ∪ ran
(𝑌‘𝑐) = ∪ ran (𝑌‘suc 𝑒)) |
13 | | eqid 2739 |
. . . . . . . . . 10
⊢ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) |
14 | | fvex 6751 |
. . . . . . . . . . . 12
⊢ (𝑌‘suc 𝑒) ∈ V |
15 | 14 | rnex 7711 |
. . . . . . . . . . 11
⊢ ran
(𝑌‘suc 𝑒) ∈ V |
16 | 15 | uniex 7550 |
. . . . . . . . . 10
⊢ ∪ ran (𝑌‘suc 𝑒) ∈ V |
17 | 12, 13, 16 | fvmpt 6839 |
. . . . . . . . 9
⊢ (suc
𝑒 ∈ ω →
((𝑐 ∈ ω ↦
∪ ran (𝑌‘𝑐))‘suc 𝑒) = ∪ ran (𝑌‘suc 𝑒)) |
18 | 9, 17 | syl 17 |
. . . . . . . 8
⊢ (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) = ∪ ran (𝑌‘suc 𝑒)) |
19 | | fveq2 6738 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑒 → (𝑌‘𝑐) = (𝑌‘𝑒)) |
20 | 19 | rneqd 5824 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘𝑒)) |
21 | 20 | unieqd 4849 |
. . . . . . . . 9
⊢ (𝑐 = 𝑒 → ∪ ran
(𝑌‘𝑐) = ∪ ran (𝑌‘𝑒)) |
22 | | fvex 6751 |
. . . . . . . . . . 11
⊢ (𝑌‘𝑒) ∈ V |
23 | 22 | rnex 7711 |
. . . . . . . . . 10
⊢ ran
(𝑌‘𝑒) ∈ V |
24 | 23 | uniex 7550 |
. . . . . . . . 9
⊢ ∪ ran (𝑌‘𝑒) ∈ V |
25 | 21, 13, 24 | fvmpt 6839 |
. . . . . . . 8
⊢ (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) = ∪ ran (𝑌‘𝑒)) |
26 | 18, 25 | sseq12d 3950 |
. . . . . . 7
⊢ (𝑒 ∈ ω → (((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) ↔ ∪ ran
(𝑌‘suc 𝑒) ⊆ ∪ ran (𝑌‘𝑒))) |
27 | 26 | adantl 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → (((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) ↔ ∪ ran
(𝑌‘suc 𝑒) ⊆ ∪ ran (𝑌‘𝑒))) |
28 | 8, 27 | mpbird 260 |
. . . . 5
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
29 | 28 | ralrimiva 3107 |
. . . 4
⊢ (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
30 | 29 | adantr 484 |
. . 3
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
31 | | fveq1 6737 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒)) |
32 | | fveq1 6737 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
33 | 31, 32 | sseq12d 3950 |
. . . . . 6
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ((𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) ↔ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒))) |
34 | 33 | ralbidv 3120 |
. . . . 5
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) ↔ ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒))) |
35 | | rneq 5822 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
36 | 35 | inteqd 4880 |
. . . . . 6
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ∩ ran
𝑑 = ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
37 | 36, 35 | eleq12d 2834 |
. . . . 5
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (∩ ran
𝑑 ∈ ran 𝑑 ↔ ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))) |
38 | 34, 37 | imbi12d 348 |
. . . 4
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ((∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑) ↔ (∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) → ∩ ran
(𝑐 ∈ ω ↦
∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))))) |
39 | 1 | isfin3ds 9970 |
. . . . . 6
⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺 ↑m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑))) |
40 | 39 | ibi 270 |
. . . . 5
⊢ (𝐺 ∈ 𝐹 → ∀𝑑 ∈ (𝒫 𝐺 ↑m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑)) |
41 | 40 | adantl 485 |
. . . 4
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∀𝑑 ∈ (𝒫 𝐺 ↑m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑)) |
42 | 1, 2, 3, 4, 5 | fin23lem34 9987 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ω) → ((𝑌‘𝑐):ω–1-1→V ∧ ∪ ran (𝑌‘𝑐) ⊆ 𝐺)) |
43 | 42 | simprd 499 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ⊆ 𝐺) |
44 | 43 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ⊆ 𝐺) |
45 | | elpw2g 5253 |
. . . . . . . 8
⊢ (𝐺 ∈ 𝐹 → (∪ ran
(𝑌‘𝑐) ∈ 𝒫 𝐺 ↔ ∪ ran
(𝑌‘𝑐) ⊆ 𝐺)) |
46 | 45 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → (∪ ran (𝑌‘𝑐) ∈ 𝒫 𝐺 ↔ ∪ ran
(𝑌‘𝑐) ⊆ 𝐺)) |
47 | 44, 46 | mpbird 260 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ∈ 𝒫 𝐺) |
48 | 47 | fmpttd 6953 |
. . . . 5
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺) |
49 | | pwexg 5287 |
. . . . . 6
⊢ (𝐺 ∈ 𝐹 → 𝒫 𝐺 ∈ V) |
50 | | vex 3426 |
. . . . . . . 8
⊢ ℎ ∈ V |
51 | | f1f 6636 |
. . . . . . . 8
⊢ (ℎ:ω–1-1→V → ℎ:ω⟶V) |
52 | | dmfex 7706 |
. . . . . . . 8
⊢ ((ℎ ∈ V ∧ ℎ:ω⟶V) → ω
∈ V) |
53 | 50, 51, 52 | sylancr 590 |
. . . . . . 7
⊢ (ℎ:ω–1-1→V → ω ∈ V) |
54 | 2, 53 | syl 17 |
. . . . . 6
⊢ (𝜑 → ω ∈
V) |
55 | | elmapg 8544 |
. . . . . 6
⊢
((𝒫 𝐺 ∈
V ∧ ω ∈ V) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑m ω) ↔ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺)) |
56 | 49, 54, 55 | syl2anr 600 |
. . . . 5
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑m ω) ↔ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺)) |
57 | 48, 56 | mpbird 260 |
. . . 4
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑m
ω)) |
58 | 38, 41, 57 | rspcdva 3553 |
. . 3
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) → ∩ ran
(𝑐 ∈ ω ↦
∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))) |
59 | 30, 58 | mpd 15 |
. 2
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∩ ran
(𝑐 ∈ ω ↦
∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
60 | 6, 59 | mtand 816 |
1
⊢ (𝜑 → ¬ 𝐺 ∈ 𝐹) |