Step | Hyp | Ref
| Expression |
1 | | epweon 7560 |
. . . . 5
⊢ E We
On |
2 | | fveq2 6717 |
. . . . . . . . . . 11
⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅)) |
3 | 2 | eleq1d 2822 |
. . . . . . . . . 10
⊢ (𝑦 = ∅ → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘∅) ∈ On)) |
4 | | fveq2 6717 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) |
5 | 4 | eleq1d 2822 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘𝑧) ∈ On)) |
6 | | fveq2 6717 |
. . . . . . . . . . 11
⊢ (𝑦 = suc 𝑧 → (𝐹‘𝑦) = (𝐹‘suc 𝑧)) |
7 | 6 | eleq1d 2822 |
. . . . . . . . . 10
⊢ (𝑦 = suc 𝑧 → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘suc 𝑧) ∈ On)) |
8 | | simpl 486 |
. . . . . . . . . 10
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝐹‘∅) ∈ On) |
9 | | suceq 6278 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧) |
10 | 9 | fveq2d 6721 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧)) |
11 | | fveq2 6717 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) |
12 | 10, 11 | eleq12d 2832 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧))) |
13 | 12 | rspcv 3532 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ω →
(∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧))) |
14 | | onelon 6238 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑧) ∈ On ∧ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)) → (𝐹‘suc 𝑧) ∈ On) |
15 | 14 | expcom 417 |
. . . . . . . . . . . 12
⊢ ((𝐹‘suc 𝑧) ∈ (𝐹‘𝑧) → ((𝐹‘𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)) |
16 | 13, 15 | syl6 35 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ω →
(∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → ((𝐹‘𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On))) |
17 | 16 | adantld 494 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ω → (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐹‘𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On))) |
18 | 3, 5, 7, 8, 17 | finds2 7678 |
. . . . . . . . 9
⊢ (𝑦 ∈ ω → (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝐹‘𝑦) ∈ On)) |
19 | 18 | com12 32 |
. . . . . . . 8
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝑦 ∈ ω → (𝐹‘𝑦) ∈ On)) |
20 | 19 | ralrimiv 3104 |
. . . . . . 7
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∀𝑦 ∈ ω (𝐹‘𝑦) ∈ On) |
21 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑦 ∈ ω ↦ (𝐹‘𝑦)) |
22 | 21 | fmpt 6927 |
. . . . . . 7
⊢
(∀𝑦 ∈
ω (𝐹‘𝑦) ∈ On ↔ (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On) |
23 | 20, 22 | sylib 221 |
. . . . . 6
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On) |
24 | 23 | frnd 6553 |
. . . . 5
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ⊆ On) |
25 | | peano1 7667 |
. . . . . . . 8
⊢ ∅
∈ ω |
26 | 23 | fdmd 6556 |
. . . . . . . 8
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ω) |
27 | 25, 26 | eleqtrrid 2845 |
. . . . . . 7
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∅ ∈ dom (𝑦 ∈ ω ↦ (𝐹‘𝑦))) |
28 | 27 | ne0d 4250 |
. . . . . 6
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅) |
29 | | dm0rn0 5794 |
. . . . . . 7
⊢ (dom
(𝑦 ∈ ω ↦
(𝐹‘𝑦)) = ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅) |
30 | 29 | necon3bii 2993 |
. . . . . 6
⊢ (dom
(𝑦 ∈ ω ↦
(𝐹‘𝑦)) ≠ ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅) |
31 | 28, 30 | sylib 221 |
. . . . 5
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅) |
32 | | wefrc 5545 |
. . . . 5
⊢ (( E We
On ∧ ran (𝑦 ∈
ω ↦ (𝐹‘𝑦)) ⊆ On ∧ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅) |
33 | 1, 24, 31, 32 | mp3an2i 1468 |
. . . 4
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅) |
34 | | fvex 6730 |
. . . . . 6
⊢ (𝐹‘𝑤) ∈ V |
35 | 34 | rgenw 3073 |
. . . . 5
⊢
∀𝑤 ∈
ω (𝐹‘𝑤) ∈ V |
36 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤)) |
37 | 36 | cbvmptv 5158 |
. . . . . 6
⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤)) |
38 | | ineq2 4121 |
. . . . . . 7
⊢ (𝑧 = (𝐹‘𝑤) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤))) |
39 | 38 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑧 = (𝐹‘𝑤) → ((ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅ ↔ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)) |
40 | 37, 39 | rexrnmptw 6914 |
. . . . 5
⊢
(∀𝑤 ∈
ω (𝐹‘𝑤) ∈ V → (∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅ ↔ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)) |
41 | 35, 40 | ax-mp 5 |
. . . 4
⊢
(∃𝑧 ∈ ran
(𝑦 ∈ ω ↦
(𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅ ↔ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅) |
42 | 33, 41 | sylib 221 |
. . 3
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅) |
43 | | peano2 7668 |
. . . . . . . . 9
⊢ (𝑤 ∈ ω → suc 𝑤 ∈
ω) |
44 | 43 | adantl 485 |
. . . . . . . 8
⊢ ((((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → suc 𝑤 ∈
ω) |
45 | | eqid 2737 |
. . . . . . . 8
⊢ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤) |
46 | | fveq2 6717 |
. . . . . . . . 9
⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤)) |
47 | 46 | rspceeqv 3552 |
. . . . . . . 8
⊢ ((suc
𝑤 ∈ ω ∧
(𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦)) |
48 | 44, 45, 47 | sylancl 589 |
. . . . . . 7
⊢ ((((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦)) |
49 | | fvex 6730 |
. . . . . . . 8
⊢ (𝐹‘suc 𝑤) ∈ V |
50 | 21 | elrnmpt 5825 |
. . . . . . . 8
⊢ ((𝐹‘suc 𝑤) ∈ V → ((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ↔ ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))) |
51 | 49, 50 | ax-mp 5 |
. . . . . . 7
⊢ ((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ↔ ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦)) |
52 | 48, 51 | sylibr 237 |
. . . . . 6
⊢ ((((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))) |
53 | | suceq 6278 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤) |
54 | 53 | fveq2d 6721 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤)) |
55 | | fveq2 6717 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) |
56 | 54, 55 | eleq12d 2832 |
. . . . . . . 8
⊢ (𝑥 = 𝑤 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤))) |
57 | 56 | rspccva 3536 |
. . . . . . 7
⊢
((∀𝑥 ∈
ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤)) |
58 | 57 | adantll 714 |
. . . . . 6
⊢ ((((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤)) |
59 | | inelcm 4379 |
. . . . . 6
⊢ (((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∧ (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤)) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) ≠ ∅) |
60 | 52, 58, 59 | syl2anc 587 |
. . . . 5
⊢ ((((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) ≠ ∅) |
61 | 60 | neneqd 2945 |
. . . 4
⊢ ((((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ¬ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅) |
62 | 61 | nrexdv 3189 |
. . 3
⊢ (((𝐹‘∅) ∈ On ∧
∀𝑥 ∈ ω
(𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅) |
63 | 42, 62 | pm2.65da 817 |
. 2
⊢ ((𝐹‘∅) ∈ On →
¬ ∀𝑥 ∈
ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) |
64 | | rexnal 3160 |
. 2
⊢
(∃𝑥 ∈
ω ¬ (𝐹‘suc
𝑥) ∈ (𝐹‘𝑥) ↔ ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) |
65 | 63, 64 | sylibr 237 |
1
⊢ ((𝐹‘∅) ∈ On →
∃𝑥 ∈ ω
¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) |