Proof of Theorem indpi
Step | Hyp | Ref
| Expression |
1 | | indpi.4 |
. 2
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
2 | | elni 7223 |
. . 3
⊢ (𝑥 ∈ N ↔
(𝑥 ∈ ω ∧
𝑥 ≠
∅)) |
3 | | eqid 2157 |
. . . . . . . . . 10
⊢ ∅ =
∅ |
4 | 3 | orci 721 |
. . . . . . . . 9
⊢ (∅
= ∅ ∨ [∅ / 𝑥]𝜑) |
5 | | nfv 1508 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥∅ =
∅ |
6 | | nfsbc1v 2955 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥[∅ / 𝑥]𝜑 |
7 | 5, 6 | nfor 1554 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(∅ =
∅ ∨ [∅ / 𝑥]𝜑) |
8 | | 0ex 4091 |
. . . . . . . . . 10
⊢ ∅
∈ V |
9 | | eqeq1 2164 |
. . . . . . . . . . 11
⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ =
∅)) |
10 | | sbceq1a 2946 |
. . . . . . . . . . 11
⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑)) |
11 | 9, 10 | orbi12d 783 |
. . . . . . . . . 10
⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨
[∅ / 𝑥]𝜑))) |
12 | 7, 8, 11 | elabf 2855 |
. . . . . . . . 9
⊢ (∅
∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨
[∅ / 𝑥]𝜑)) |
13 | 4, 12 | mpbir 145 |
. . . . . . . 8
⊢ ∅
∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} |
14 | | suceq 4362 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅) |
15 | | df-1o 6360 |
. . . . . . . . . . . . . 14
⊢
1o = suc ∅ |
16 | 14, 15 | eqtr4di 2208 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ∅ → suc 𝑦 =
1o) |
17 | | indpi.5 |
. . . . . . . . . . . . . . 15
⊢ 𝜓 |
18 | 17 | olci 722 |
. . . . . . . . . . . . . 14
⊢
(1o = ∅ ∨ 𝜓) |
19 | | 1oex 6368 |
. . . . . . . . . . . . . . 15
⊢
1o ∈ V |
20 | | eqeq1 2164 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 1o → (𝑥 = ∅ ↔ 1o
= ∅)) |
21 | | indpi.1 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 1o → (𝜑 ↔ 𝜓)) |
22 | 20, 21 | orbi12d 783 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 1o → ((𝑥 = ∅ ∨ 𝜑) ↔ (1o = ∅ ∨ 𝜓))) |
23 | 19, 22 | elab 2856 |
. . . . . . . . . . . . . 14
⊢
(1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1o = ∅ ∨
𝜓)) |
24 | 18, 23 | mpbir 145 |
. . . . . . . . . . . . 13
⊢
1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} |
25 | 16, 24 | eqeltrdi 2248 |
. . . . . . . . . . . 12
⊢ (𝑦 = ∅ → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}) |
26 | 25 | a1d 22 |
. . . . . . . . . . 11
⊢ (𝑦 = ∅ → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) |
27 | 26 | a1i 9 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ω → (𝑦 = ∅ → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))) |
28 | | indpi.6 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ N →
(𝜒 → 𝜃)) |
29 | | elni 7223 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ N ↔
(𝑦 ∈ ω ∧
𝑦 ≠
∅)) |
30 | 29 | simprbi 273 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ N →
𝑦 ≠
∅) |
31 | 30 | neneqd 2348 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ N →
¬ 𝑦 =
∅) |
32 | | biorf 734 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒))) |
33 | 31, 32 | syl 14 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ N →
(𝜒 ↔ (𝑦 = ∅ ∨ 𝜒))) |
34 | | vex 2715 |
. . . . . . . . . . . . . 14
⊢ 𝑦 ∈ V |
35 | | eqeq1 2164 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) |
36 | | indpi.2 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
37 | 35, 36 | orbi12d 783 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒))) |
38 | 34, 37 | elab 2856 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒)) |
39 | 33, 38 | bitr4di 197 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ N →
(𝜒 ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) |
40 | | 1pi 7230 |
. . . . . . . . . . . . . . . . . 18
⊢
1o ∈ N |
41 | | addclpi 7242 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 ∈ N ∧
1o ∈ N) → (𝑦 +N 1o)
∈ N) |
42 | 40, 41 | mpan2 422 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ N →
(𝑦
+N 1o) ∈
N) |
43 | | elni 7223 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑦 +N
1o) ∈ N ↔ ((𝑦 +N 1o)
∈ ω ∧ (𝑦
+N 1o) ≠ ∅)) |
44 | 42, 43 | sylib 121 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ N →
((𝑦
+N 1o) ∈ ω ∧ (𝑦 +N
1o) ≠ ∅)) |
45 | 44 | simprd 113 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ N →
(𝑦
+N 1o) ≠ ∅) |
46 | 45 | neneqd 2348 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ N →
¬ (𝑦
+N 1o) = ∅) |
47 | | biorf 734 |
. . . . . . . . . . . . . 14
⊢ (¬
(𝑦
+N 1o) = ∅ → (𝜃 ↔ ((𝑦 +N 1o) =
∅ ∨ 𝜃))) |
48 | 46, 47 | syl 14 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ N →
(𝜃 ↔ ((𝑦 +N
1o) = ∅ ∨ 𝜃))) |
49 | | eqeq1 2164 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 +N 1o)
→ (𝑥 = ∅ ↔
(𝑦
+N 1o) = ∅)) |
50 | | indpi.3 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 +N 1o)
→ (𝜑 ↔ 𝜃)) |
51 | 49, 50 | orbi12d 783 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑦 +N 1o)
→ ((𝑥 = ∅ ∨
𝜑) ↔ ((𝑦 +N
1o) = ∅ ∨ 𝜃))) |
52 | 51 | elabg 2858 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 +N
1o) ∈ N → ((𝑦 +N 1o)
∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ((𝑦 +N 1o) =
∅ ∨ 𝜃))) |
53 | 42, 52 | syl 14 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ N →
((𝑦
+N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ((𝑦 +N 1o) =
∅ ∨ 𝜃))) |
54 | | addpiord 7231 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ N ∧
1o ∈ N) → (𝑦 +N 1o) =
(𝑦 +o
1o)) |
55 | 40, 54 | mpan2 422 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ N →
(𝑦
+N 1o) = (𝑦 +o
1o)) |
56 | | pion 7225 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ N →
𝑦 ∈
On) |
57 | | oa1suc 6411 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ On → (𝑦 +o 1o) =
suc 𝑦) |
58 | 56, 57 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ N →
(𝑦 +o
1o) = suc 𝑦) |
59 | 55, 58 | eqtrd 2190 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ N →
(𝑦
+N 1o) = suc 𝑦) |
60 | 59 | eleq1d 2226 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ N →
((𝑦
+N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) |
61 | 48, 53, 60 | 3bitr2d 215 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ N →
(𝜃 ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) |
62 | 28, 39, 61 | 3imtr3d 201 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ N →
(𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) |
63 | 62 | a1i 9 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ω → (𝑦 ∈ N →
(𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))) |
64 | | nndceq0 4576 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ω →
DECID 𝑦 =
∅) |
65 | | df-dc 821 |
. . . . . . . . . . . 12
⊢
(DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅)) |
66 | 64, 65 | sylib 121 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅)) |
67 | | idd 21 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅)) |
68 | 67 | necon3bd 2370 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ω → (¬
𝑦 = ∅ → 𝑦 ≠ ∅)) |
69 | 68 | anc2li 327 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ω → (¬
𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠
∅))) |
70 | 69, 29 | syl6ibr 161 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ω → (¬
𝑦 = ∅ → 𝑦 ∈
N)) |
71 | 70 | orim2d 778 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦 ∈
N))) |
72 | 66, 71 | mpd 13 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦 ∈
N)) |
73 | 27, 63, 72 | mpjaod 708 |
. . . . . . . . 9
⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) |
74 | 73 | rgen 2510 |
. . . . . . . 8
⊢
∀𝑦 ∈
ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}) |
75 | | peano5 4556 |
. . . . . . . 8
⊢ ((∅
∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}) |
76 | 13, 74, 75 | mp2an 423 |
. . . . . . 7
⊢ ω
⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} |
77 | 76 | sseli 3124 |
. . . . . 6
⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}) |
78 | | abid 2145 |
. . . . . 6
⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑥 = ∅ ∨ 𝜑)) |
79 | 77, 78 | sylib 121 |
. . . . 5
⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ 𝜑)) |
80 | 79 | adantr 274 |
. . . 4
⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝑥 = ∅ ∨ 𝜑)) |
81 | | df-ne 2328 |
. . . . . 6
⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅) |
82 | | biorf 734 |
. . . . . 6
⊢ (¬
𝑥 = ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑))) |
83 | 81, 82 | sylbi 120 |
. . . . 5
⊢ (𝑥 ≠ ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑))) |
84 | 83 | adantl 275 |
. . . 4
⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑))) |
85 | 80, 84 | mpbird 166 |
. . 3
⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → 𝜑) |
86 | 2, 85 | sylbi 120 |
. 2
⊢ (𝑥 ∈ N →
𝜑) |
87 | 1, 86 | vtoclga 2778 |
1
⊢ (𝐴 ∈ N →
𝜏) |