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 9171 |
. 2
⊢ (𝜑 → ¬ ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
7 | 1, 2, 3, 4, 5 | fin23lem34 9168 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ω) → ((𝑌‘𝑐):ω–1-1→V ∧ ∪ ran (𝑌‘𝑐) ⊆ 𝐺)) |
8 | 7 | simprd 479 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ⊆ 𝐺) |
9 | 8 | adantlr 751 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ⊆ 𝐺) |
10 | | elpw2g 4827 |
. . . . . . 7
⊢ (𝐺 ∈ 𝐹 → (∪ ran
(𝑌‘𝑐) ∈ 𝒫 𝐺 ↔ ∪ ran
(𝑌‘𝑐) ⊆ 𝐺)) |
11 | 10 | ad2antlr 763 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → (∪ ran (𝑌‘𝑐) ∈ 𝒫 𝐺 ↔ ∪ ran
(𝑌‘𝑐) ⊆ 𝐺)) |
12 | 9, 11 | mpbird 247 |
. . . . 5
⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ∈ 𝒫 𝐺) |
13 | | eqid 2622 |
. . . . 5
⊢ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) |
14 | 12, 13 | fmptd 6385 |
. . . 4
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺) |
15 | | pwexg 4850 |
. . . . 5
⊢ (𝐺 ∈ 𝐹 → 𝒫 𝐺 ∈ V) |
16 | | vex 3203 |
. . . . . . 7
⊢ ℎ ∈ V |
17 | | f1f 6101 |
. . . . . . 7
⊢ (ℎ:ω–1-1→V → ℎ:ω⟶V) |
18 | | dmfex 7124 |
. . . . . . 7
⊢ ((ℎ ∈ V ∧ ℎ:ω⟶V) → ω
∈ V) |
19 | 16, 17, 18 | sylancr 695 |
. . . . . 6
⊢ (ℎ:ω–1-1→V → ω ∈ V) |
20 | 2, 19 | syl 17 |
. . . . 5
⊢ (𝜑 → ω ∈
V) |
21 | | elmapg 7870 |
. . . . 5
⊢
((𝒫 𝐺 ∈
V ∧ ω ∈ V) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑𝑚 ω) ↔
(𝑐 ∈ ω ↦
∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺)) |
22 | 15, 20, 21 | syl2anr 495 |
. . . 4
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑𝑚 ω) ↔
(𝑐 ∈ ω ↦
∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺)) |
23 | 14, 22 | mpbird 247 |
. . 3
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑𝑚
ω)) |
24 | 1 | isfin3ds 9151 |
. . . . 5
⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺 ↑𝑚
ω)(∀𝑒 ∈
ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑))) |
25 | 24 | ibi 256 |
. . . 4
⊢ (𝐺 ∈ 𝐹 → ∀𝑑 ∈ (𝒫 𝐺 ↑𝑚
ω)(∀𝑒 ∈
ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑)) |
26 | 25 | adantl 482 |
. . 3
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∀𝑑 ∈ (𝒫 𝐺 ↑𝑚
ω)(∀𝑒 ∈
ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑)) |
27 | 1, 2, 3, 4, 5 | fin23lem35 9169 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊊ ∪ ran
(𝑌‘𝑒)) |
28 | 27 | pssssd 3704 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊆ ∪ ran
(𝑌‘𝑒)) |
29 | | peano2 7086 |
. . . . . . . . 9
⊢ (𝑒 ∈ ω → suc 𝑒 ∈
ω) |
30 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑐 = suc 𝑒 → (𝑌‘𝑐) = (𝑌‘suc 𝑒)) |
31 | 30 | rneqd 5353 |
. . . . . . . . . . 11
⊢ (𝑐 = suc 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘suc 𝑒)) |
32 | 31 | unieqd 4446 |
. . . . . . . . . 10
⊢ (𝑐 = suc 𝑒 → ∪ ran
(𝑌‘𝑐) = ∪ ran (𝑌‘suc 𝑒)) |
33 | | fvex 6201 |
. . . . . . . . . . . 12
⊢ (𝑌‘suc 𝑒) ∈ V |
34 | 33 | rnex 7100 |
. . . . . . . . . . 11
⊢ ran
(𝑌‘suc 𝑒) ∈ V |
35 | 34 | uniex 6953 |
. . . . . . . . . 10
⊢ ∪ ran (𝑌‘suc 𝑒) ∈ V |
36 | 32, 13, 35 | fvmpt 6282 |
. . . . . . . . 9
⊢ (suc
𝑒 ∈ ω →
((𝑐 ∈ ω ↦
∪ ran (𝑌‘𝑐))‘suc 𝑒) = ∪ ran (𝑌‘suc 𝑒)) |
37 | 29, 36 | syl 17 |
. . . . . . . 8
⊢ (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) = ∪ ran (𝑌‘suc 𝑒)) |
38 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑒 → (𝑌‘𝑐) = (𝑌‘𝑒)) |
39 | 38 | rneqd 5353 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘𝑒)) |
40 | 39 | unieqd 4446 |
. . . . . . . . 9
⊢ (𝑐 = 𝑒 → ∪ ran
(𝑌‘𝑐) = ∪ ran (𝑌‘𝑒)) |
41 | | fvex 6201 |
. . . . . . . . . . 11
⊢ (𝑌‘𝑒) ∈ V |
42 | 41 | rnex 7100 |
. . . . . . . . . 10
⊢ ran
(𝑌‘𝑒) ∈ V |
43 | 42 | uniex 6953 |
. . . . . . . . 9
⊢ ∪ ran (𝑌‘𝑒) ∈ V |
44 | 40, 13, 43 | fvmpt 6282 |
. . . . . . . 8
⊢ (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) = ∪ ran (𝑌‘𝑒)) |
45 | 37, 44 | sseq12d 3634 |
. . . . . . 7
⊢ (𝑒 ∈ ω → (((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) ↔ ∪ ran
(𝑌‘suc 𝑒) ⊆ ∪ ran (𝑌‘𝑒))) |
46 | 45 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → (((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) ↔ ∪ ran
(𝑌‘suc 𝑒) ⊆ ∪ ran (𝑌‘𝑒))) |
47 | 28, 46 | mpbird 247 |
. . . . 5
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
48 | 47 | ralrimiva 2966 |
. . . 4
⊢ (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
49 | 48 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
50 | | fveq1 6190 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒)) |
51 | | fveq1 6190 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
52 | 50, 51 | sseq12d 3634 |
. . . . . 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))) |
53 | 52 | ralbidv 2986 |
. . . . 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))) |
54 | | rneq 5351 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
55 | 54 | inteqd 4480 |
. . . . . 6
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ∩ ran
𝑑 = ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
56 | 55, 54 | eleq12d 2695 |
. . . . 5
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (∩ ran
𝑑 ∈ ran 𝑑 ↔ ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))) |
57 | 53, 56 | imbi12d 334 |
. . . 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))))) |
58 | 57 | rspcv 3305 |
. . 3
⊢ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑𝑚 ω) →
(∀𝑑 ∈
(𝒫 𝐺
↑𝑚 ω)(∀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))))) |
59 | 23, 26, 49, 58 | syl3c 66 |
. 2
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∩ ran
(𝑐 ∈ ω ↦
∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
60 | 6, 59 | mtand 691 |
1
⊢ (𝜑 → ¬ 𝐺 ∈ 𝐹) |