| 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 10389 |
. 2
⊢ (𝜑 → ¬ ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
| 7 | 1, 2, 3, 4, 5 | fin23lem35 10387 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊊ ∪ ran
(𝑌‘𝑒)) |
| 8 | 7 | pssssd 4100 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊆ ∪ ran
(𝑌‘𝑒)) |
| 9 | | peano2 7912 |
. . . . . . . . 9
⊢ (𝑒 ∈ ω → suc 𝑒 ∈
ω) |
| 10 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑐 = suc 𝑒 → (𝑌‘𝑐) = (𝑌‘suc 𝑒)) |
| 11 | 10 | rneqd 5949 |
. . . . . . . . . . 11
⊢ (𝑐 = suc 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘suc 𝑒)) |
| 12 | 11 | unieqd 4920 |
. . . . . . . . . 10
⊢ (𝑐 = suc 𝑒 → ∪ ran
(𝑌‘𝑐) = ∪ ran (𝑌‘suc 𝑒)) |
| 13 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) |
| 14 | | fvex 6919 |
. . . . . . . . . . . 12
⊢ (𝑌‘suc 𝑒) ∈ V |
| 15 | 14 | rnex 7932 |
. . . . . . . . . . 11
⊢ ran
(𝑌‘suc 𝑒) ∈ V |
| 16 | 15 | uniex 7761 |
. . . . . . . . . 10
⊢ ∪ ran (𝑌‘suc 𝑒) ∈ V |
| 17 | 12, 13, 16 | fvmpt 7016 |
. . . . . . . . 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 6906 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑒 → (𝑌‘𝑐) = (𝑌‘𝑒)) |
| 20 | 19 | rneqd 5949 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘𝑒)) |
| 21 | 20 | unieqd 4920 |
. . . . . . . . 9
⊢ (𝑐 = 𝑒 → ∪ ran
(𝑌‘𝑐) = ∪ ran (𝑌‘𝑒)) |
| 22 | | fvex 6919 |
. . . . . . . . . . 11
⊢ (𝑌‘𝑒) ∈ V |
| 23 | 22 | rnex 7932 |
. . . . . . . . . 10
⊢ ran
(𝑌‘𝑒) ∈ V |
| 24 | 23 | uniex 7761 |
. . . . . . . . 9
⊢ ∪ ran (𝑌‘𝑒) ∈ V |
| 25 | 21, 13, 24 | fvmpt 7016 |
. . . . . . . 8
⊢ (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) = ∪ ran (𝑌‘𝑒)) |
| 26 | 18, 25 | sseq12d 4017 |
. . . . . . 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 481 |
. . . . . 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 257 |
. . . . 5
⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
| 29 | 28 | ralrimiva 3146 |
. . . 4
⊢ (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
| 30 | 29 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
| 31 | | fveq1 6905 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒)) |
| 32 | | fveq1 6905 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)) |
| 33 | 31, 32 | sseq12d 4017 |
. . . . . 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 3178 |
. . . . 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 5947 |
. . . . . . 7
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
| 36 | 35 | inteqd 4951 |
. . . . . 6
⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ∩ ran
𝑑 = ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))) |
| 37 | 36, 35 | eleq12d 2835 |
. . . . 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 344 |
. . . 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 10369 |
. . . . . 6
⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺 ↑m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑))) |
| 40 | 39 | ibi 267 |
. . . . 5
⊢ (𝐺 ∈ 𝐹 → ∀𝑑 ∈ (𝒫 𝐺 ↑m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran
𝑑 ∈ ran 𝑑)) |
| 41 | 40 | adantl 481 |
. . . 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 10386 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ω) → ((𝑌‘𝑐):ω–1-1→V ∧ ∪ ran (𝑌‘𝑐) ⊆ 𝐺)) |
| 43 | 42 | simprd 495 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ⊆ 𝐺) |
| 44 | 43 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ⊆ 𝐺) |
| 45 | | elpw2g 5333 |
. . . . . . . 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 257 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ∈ 𝒫 𝐺) |
| 48 | 47 | fmpttd 7135 |
. . . . 5
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺) |
| 49 | | pwexg 5378 |
. . . . . 6
⊢ (𝐺 ∈ 𝐹 → 𝒫 𝐺 ∈ V) |
| 50 | | vex 3484 |
. . . . . . . 8
⊢ ℎ ∈ V |
| 51 | | f1f 6804 |
. . . . . . . 8
⊢ (ℎ:ω–1-1→V → ℎ:ω⟶V) |
| 52 | | dmfex 7927 |
. . . . . . . 8
⊢ ((ℎ ∈ V ∧ ℎ:ω⟶V) → ω
∈ V) |
| 53 | 50, 51, 52 | sylancr 587 |
. . . . . . 7
⊢ (ℎ:ω–1-1→V → ω ∈ V) |
| 54 | 2, 53 | syl 17 |
. . . . . 6
⊢ (𝜑 → ω ∈
V) |
| 55 | | elmapg 8879 |
. . . . . 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 597 |
. . . . 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 257 |
. . . 4
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑m
ω)) |
| 58 | 38, 41, 57 | rspcdva 3623 |
. . 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) |