Proof of Theorem indpi
Step | Hyp | Ref
| Expression |
1 | | 1pi 9705 |
. . . . . . 7
⊢
1𝑜 ∈ N |
2 | 1 | elexi 3213 |
. . . . . 6
⊢
1𝑜 ∈ V |
3 | 2 | eqvinc 3330 |
. . . . 5
⊢
(1𝑜 = 𝐴 ↔ ∃𝑥(𝑥 = 1𝑜 ∧ 𝑥 = 𝐴)) |
4 | | indpi.4 |
. . . . 5
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
5 | | indpi.5 |
. . . . . 6
⊢ 𝜓 |
6 | | indpi.1 |
. . . . . 6
⊢ (𝑥 = 1𝑜 →
(𝜑 ↔ 𝜓)) |
7 | 5, 6 | mpbiri 248 |
. . . . 5
⊢ (𝑥 = 1𝑜 →
𝜑) |
8 | 3, 4, 7 | gencl 3235 |
. . . 4
⊢
(1𝑜 = 𝐴 → 𝜏) |
9 | 8 | eqcoms 2630 |
. . 3
⊢ (𝐴 = 1𝑜 →
𝜏) |
10 | 9 | a1i 11 |
. 2
⊢ (𝐴 ∈ N →
(𝐴 = 1𝑜
→ 𝜏)) |
11 | | pinn 9700 |
. . . . 5
⊢ (𝐴 ∈ N →
𝐴 ∈
ω) |
12 | | elni2 9699 |
. . . . . 6
⊢ (𝐴 ∈ N ↔
(𝐴 ∈ ω ∧
∅ ∈ 𝐴)) |
13 | | nnord 7073 |
. . . . . . . . 9
⊢ (𝐴 ∈ ω → Ord 𝐴) |
14 | | ordsucss 7018 |
. . . . . . . . 9
⊢ (Ord
𝐴 → (∅ ∈
𝐴 → suc ∅
⊆ 𝐴)) |
15 | 13, 14 | syl 17 |
. . . . . . . 8
⊢ (𝐴 ∈ ω → (∅
∈ 𝐴 → suc ∅
⊆ 𝐴)) |
16 | | df-1o 7560 |
. . . . . . . . 9
⊢
1𝑜 = suc ∅ |
17 | 16 | sseq1i 3629 |
. . . . . . . 8
⊢
(1𝑜 ⊆ 𝐴 ↔ suc ∅ ⊆ 𝐴) |
18 | 15, 17 | syl6ibr 242 |
. . . . . . 7
⊢ (𝐴 ∈ ω → (∅
∈ 𝐴 →
1𝑜 ⊆ 𝐴)) |
19 | 18 | imp 445 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧ ∅
∈ 𝐴) →
1𝑜 ⊆ 𝐴) |
20 | 12, 19 | sylbi 207 |
. . . . 5
⊢ (𝐴 ∈ N →
1𝑜 ⊆ 𝐴) |
21 | | 1onn 7719 |
. . . . . 6
⊢
1𝑜 ∈ ω |
22 | | eleq1 2689 |
. . . . . . . . 9
⊢ (𝑥 = 1𝑜 →
(𝑥 ∈ N
↔ 1𝑜 ∈ N)) |
23 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑥 = 1𝑜 →
(1𝑜 <N 𝑥 ↔ 1𝑜
<N 1𝑜)) |
24 | 22, 23 | anbi12d 747 |
. . . . . . . 8
⊢ (𝑥 = 1𝑜 →
((𝑥 ∈ N
∧ 1𝑜 <N 𝑥) ↔ (1𝑜 ∈
N ∧ 1𝑜 <N
1𝑜))) |
25 | 24, 6 | imbi12d 334 |
. . . . . . 7
⊢ (𝑥 = 1𝑜 →
(((𝑥 ∈ N
∧ 1𝑜 <N 𝑥) → 𝜑) ↔ ((1𝑜 ∈
N ∧ 1𝑜 <N
1𝑜) → 𝜓))) |
26 | | eleq1 2689 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 ∈ N ↔ 𝑦 ∈
N)) |
27 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (1𝑜
<N 𝑥 ↔ 1𝑜
<N 𝑦)) |
28 | 26, 27 | anbi12d 747 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) ↔ (𝑦 ∈ N ∧
1𝑜 <N 𝑦))) |
29 | | indpi.2 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
30 | 28, 29 | imbi12d 334 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 𝜑) ↔ ((𝑦 ∈ N ∧
1𝑜 <N 𝑦) → 𝜒))) |
31 | | pinn 9700 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ N →
𝑥 ∈
ω) |
32 | | eleq1 2689 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ suc 𝑦 ∈ ω)) |
33 | | peano2b 7081 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ω ↔ suc 𝑦 ∈
ω) |
34 | 32, 33 | syl6bbr 278 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω)) |
35 | 31, 34 | syl5ib 234 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ N → 𝑦 ∈
ω)) |
36 | 35 | adantrd 484 |
. . . . . . . . . . . . 13
⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 𝑦 ∈ ω)) |
37 | | ltpiord 9709 |
. . . . . . . . . . . . . . . 16
⊢
((1𝑜 ∈ N ∧ 𝑥 ∈ N) →
(1𝑜 <N 𝑥 ↔ 1𝑜 ∈ 𝑥)) |
38 | 1, 37 | mpan 706 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ N →
(1𝑜 <N 𝑥 ↔ 1𝑜 ∈ 𝑥)) |
39 | 38 | biimpa 501 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 1𝑜 ∈ 𝑥) |
40 | | eleq2 2690 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = suc 𝑦 → (1𝑜 ∈ 𝑥 ↔ 1𝑜
∈ suc 𝑦)) |
41 | | elsuci 5791 |
. . . . . . . . . . . . . . . 16
⊢
(1𝑜 ∈ suc 𝑦 → (1𝑜 ∈ 𝑦 ∨ 1𝑜 =
𝑦)) |
42 | | ne0i 3921 |
. . . . . . . . . . . . . . . . 17
⊢
(1𝑜 ∈ 𝑦 → 𝑦 ≠ ∅) |
43 | | 0lt1o 7584 |
. . . . . . . . . . . . . . . . . . 19
⊢ ∅
∈ 1𝑜 |
44 | | eleq2 2690 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1𝑜 = 𝑦 → (∅ ∈ 1𝑜
↔ ∅ ∈ 𝑦)) |
45 | 43, 44 | mpbii 223 |
. . . . . . . . . . . . . . . . . 18
⊢
(1𝑜 = 𝑦 → ∅ ∈ 𝑦) |
46 | | ne0i 3921 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∈ 𝑦 → 𝑦 ≠ ∅) |
47 | 45, 46 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢
(1𝑜 = 𝑦 → 𝑦 ≠ ∅) |
48 | 42, 47 | jaoi 394 |
. . . . . . . . . . . . . . . 16
⊢
((1𝑜 ∈ 𝑦 ∨ 1𝑜 = 𝑦) → 𝑦 ≠ ∅) |
49 | 41, 48 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
(1𝑜 ∈ suc 𝑦 → 𝑦 ≠ ∅) |
50 | 40, 49 | syl6bi 243 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = suc 𝑦 → (1𝑜 ∈ 𝑥 → 𝑦 ≠ ∅)) |
51 | 39, 50 | syl5 34 |
. . . . . . . . . . . . 13
⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 𝑦 ≠ ∅)) |
52 | 36, 51 | jcad 555 |
. . . . . . . . . . . 12
⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))) |
53 | | elni 9698 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ N ↔
(𝑦 ∈ ω ∧
𝑦 ≠
∅)) |
54 | 52, 53 | syl6ibr 242 |
. . . . . . . . . . 11
⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 𝑦 ∈ N)) |
55 | | simpr 477 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 1𝑜
<N 𝑥) |
56 | | breq2 4657 |
. . . . . . . . . . . 12
⊢ (𝑥 = suc 𝑦 → (1𝑜
<N 𝑥 ↔ 1𝑜
<N suc 𝑦)) |
57 | 55, 56 | syl5ib 234 |
. . . . . . . . . . 11
⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 1𝑜
<N suc 𝑦)) |
58 | 54, 57 | jcad 555 |
. . . . . . . . . 10
⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → (𝑦 ∈ N ∧
1𝑜 <N suc 𝑦))) |
59 | | addclpi 9714 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ N ∧
1𝑜 ∈ N) → (𝑦 +N
1𝑜) ∈ N) |
60 | 1, 59 | mpan2 707 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ N →
(𝑦
+N 1𝑜) ∈
N) |
61 | | addpiord 9706 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ N ∧
1𝑜 ∈ N) → (𝑦 +N
1𝑜) = (𝑦
+𝑜 1𝑜)) |
62 | 1, 61 | mpan2 707 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ N →
(𝑦
+N 1𝑜) = (𝑦 +𝑜
1𝑜)) |
63 | | pion 9701 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ N →
𝑦 ∈
On) |
64 | | oa1suc 7611 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ On → (𝑦 +𝑜
1𝑜) = suc 𝑦) |
65 | 63, 64 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ N →
(𝑦 +𝑜
1𝑜) = suc 𝑦) |
66 | 62, 65 | eqtrd 2656 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ N →
(𝑦
+N 1𝑜) = suc 𝑦) |
67 | 66 | eqeq2d 2632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ N →
(𝑥 = (𝑦 +N
1𝑜) ↔ 𝑥 = suc 𝑦)) |
68 | 67 | biimparc 504 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 = suc 𝑦 ∧ 𝑦 ∈ N) → 𝑥 = (𝑦 +N
1𝑜)) |
69 | 68 | eleq1d 2686 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = suc 𝑦 ∧ 𝑦 ∈ N) → (𝑥 ∈ N ↔
(𝑦
+N 1𝑜) ∈
N)) |
70 | 60, 69 | syl5ibr 236 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = suc 𝑦 ∧ 𝑦 ∈ N) → (𝑦 ∈ N →
𝑥 ∈
N)) |
71 | 70 | ex 450 |
. . . . . . . . . . . 12
⊢ (𝑥 = suc 𝑦 → (𝑦 ∈ N → (𝑦 ∈ N →
𝑥 ∈
N))) |
72 | 71 | pm2.43d 53 |
. . . . . . . . . . 11
⊢ (𝑥 = suc 𝑦 → (𝑦 ∈ N → 𝑥 ∈
N)) |
73 | 56 | biimprd 238 |
. . . . . . . . . . 11
⊢ (𝑥 = suc 𝑦 → (1𝑜
<N suc 𝑦 → 1𝑜
<N 𝑥)) |
74 | 72, 73 | anim12d 586 |
. . . . . . . . . 10
⊢ (𝑥 = suc 𝑦 → ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → (𝑥 ∈ N ∧
1𝑜 <N 𝑥))) |
75 | 58, 74 | impbid 202 |
. . . . . . . . 9
⊢ (𝑥 = suc 𝑦 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) ↔ (𝑦 ∈ N ∧
1𝑜 <N suc 𝑦))) |
76 | 75 | imbi1d 331 |
. . . . . . . 8
⊢ (𝑥 = suc 𝑦 → (((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 𝜑) ↔ ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → 𝜑))) |
77 | | indpi.3 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑦 +N
1𝑜) → (𝜑 ↔ 𝜃)) |
78 | 67, 77 | syl6bir 244 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ N →
(𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))) |
79 | 78 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))) |
80 | 79 | com12 32 |
. . . . . . . . 9
⊢ (𝑥 = suc 𝑦 → ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → (𝜑 ↔ 𝜃))) |
81 | 80 | pm5.74d 262 |
. . . . . . . 8
⊢ (𝑥 = suc 𝑦 → (((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → 𝜑) ↔ ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → 𝜃))) |
82 | 76, 81 | bitrd 268 |
. . . . . . 7
⊢ (𝑥 = suc 𝑦 → (((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 𝜑) ↔ ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → 𝜃))) |
83 | | eleq1 2689 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → (𝑥 ∈ N ↔ 𝐴 ∈
N)) |
84 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → (1𝑜
<N 𝑥 ↔ 1𝑜
<N 𝐴)) |
85 | 83, 84 | anbi12d 747 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → ((𝑥 ∈ N ∧
1𝑜 <N 𝑥) ↔ (𝐴 ∈ N ∧
1𝑜 <N 𝐴))) |
86 | 85, 4 | imbi12d 334 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → (((𝑥 ∈ N ∧
1𝑜 <N 𝑥) → 𝜑) ↔ ((𝐴 ∈ N ∧
1𝑜 <N 𝐴) → 𝜏))) |
87 | 5 | 2a1i 12 |
. . . . . . 7
⊢
(1𝑜 ∈ ω → ((1𝑜
∈ N ∧ 1𝑜
<N 1𝑜) → 𝜓)) |
88 | | ltpiord 9709 |
. . . . . . . . . . . . . . 15
⊢
((1𝑜 ∈ N ∧ 𝑦 ∈ N) →
(1𝑜 <N 𝑦 ↔ 1𝑜 ∈ 𝑦)) |
89 | 1, 88 | mpan 706 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ N →
(1𝑜 <N 𝑦 ↔ 1𝑜 ∈ 𝑦)) |
90 | 89 | pm5.32i 669 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ N ∧
1𝑜 <N 𝑦) ↔ (𝑦 ∈ N ∧
1𝑜 ∈ 𝑦)) |
91 | 90 | simplbi2 655 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ N →
(1𝑜 ∈ 𝑦 → (𝑦 ∈ N ∧
1𝑜 <N 𝑦))) |
92 | 91 | imim1d 82 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ N →
(((𝑦 ∈ N
∧ 1𝑜 <N 𝑦) → 𝜒) → (1𝑜 ∈ 𝑦 → 𝜒))) |
93 | | ltrelpi 9711 |
. . . . . . . . . . . . . . 15
⊢
<N ⊆ (N ×
N) |
94 | 93 | brel 5168 |
. . . . . . . . . . . . . 14
⊢
(1𝑜 <N suc 𝑦 → (1𝑜
∈ N ∧ suc 𝑦 ∈ N)) |
95 | | ltpiord 9709 |
. . . . . . . . . . . . . 14
⊢
((1𝑜 ∈ N ∧ suc 𝑦 ∈ N) →
(1𝑜 <N suc 𝑦 ↔ 1𝑜 ∈ suc
𝑦)) |
96 | 94, 95 | syl 17 |
. . . . . . . . . . . . 13
⊢
(1𝑜 <N suc 𝑦 → (1𝑜
<N suc 𝑦 ↔ 1𝑜 ∈ suc
𝑦)) |
97 | 96 | ibi 256 |
. . . . . . . . . . . 12
⊢
(1𝑜 <N suc 𝑦 → 1𝑜
∈ suc 𝑦) |
98 | 2 | eqvinc 3330 |
. . . . . . . . . . . . . . 15
⊢
(1𝑜 = 𝑦 ↔ ∃𝑥(𝑥 = 1𝑜 ∧ 𝑥 = 𝑦)) |
99 | 98, 29, 7 | gencl 3235 |
. . . . . . . . . . . . . 14
⊢
(1𝑜 = 𝑦 → 𝜒) |
100 | | jao 534 |
. . . . . . . . . . . . . 14
⊢
((1𝑜 ∈ 𝑦 → 𝜒) → ((1𝑜 = 𝑦 → 𝜒) → ((1𝑜 ∈
𝑦 ∨
1𝑜 = 𝑦)
→ 𝜒))) |
101 | 99, 100 | mpi 20 |
. . . . . . . . . . . . 13
⊢
((1𝑜 ∈ 𝑦 → 𝜒) → ((1𝑜 ∈
𝑦 ∨
1𝑜 = 𝑦)
→ 𝜒)) |
102 | 41, 101 | syl5 34 |
. . . . . . . . . . . 12
⊢
((1𝑜 ∈ 𝑦 → 𝜒) → (1𝑜 ∈ suc
𝑦 → 𝜒)) |
103 | 97, 102 | syl5 34 |
. . . . . . . . . . 11
⊢
((1𝑜 ∈ 𝑦 → 𝜒) → (1𝑜
<N suc 𝑦 → 𝜒)) |
104 | 92, 103 | syl6com 37 |
. . . . . . . . . 10
⊢ (((𝑦 ∈ N ∧
1𝑜 <N 𝑦) → 𝜒) → (𝑦 ∈ N →
(1𝑜 <N suc 𝑦 → 𝜒))) |
105 | 104 | impd 447 |
. . . . . . . . 9
⊢ (((𝑦 ∈ N ∧
1𝑜 <N 𝑦) → 𝜒) → ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → 𝜒)) |
106 | 16 | sseq1i 3629 |
. . . . . . . . . . 11
⊢
(1𝑜 ⊆ 𝑦 ↔ suc ∅ ⊆ 𝑦) |
107 | | 0ex 4790 |
. . . . . . . . . . . 12
⊢ ∅
∈ V |
108 | | sucssel 5819 |
. . . . . . . . . . . 12
⊢ (∅
∈ V → (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦)) |
109 | 107, 108 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (suc
∅ ⊆ 𝑦 →
∅ ∈ 𝑦) |
110 | 106, 109 | sylbi 207 |
. . . . . . . . . 10
⊢
(1𝑜 ⊆ 𝑦 → ∅ ∈ 𝑦) |
111 | | elni2 9699 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ N ↔
(𝑦 ∈ ω ∧
∅ ∈ 𝑦)) |
112 | | indpi.6 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ N →
(𝜒 → 𝜃)) |
113 | 111, 112 | sylbir 225 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ω ∧ ∅
∈ 𝑦) → (𝜒 → 𝜃)) |
114 | 110, 113 | sylan2 491 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ω ∧
1𝑜 ⊆ 𝑦) → (𝜒 → 𝜃)) |
115 | 105, 114 | syl9r 78 |
. . . . . . . 8
⊢ ((𝑦 ∈ ω ∧
1𝑜 ⊆ 𝑦) → (((𝑦 ∈ N ∧
1𝑜 <N 𝑦) → 𝜒) → ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → 𝜃))) |
116 | 115 | adantlr 751 |
. . . . . . 7
⊢ (((𝑦 ∈ ω ∧
1𝑜 ∈ ω) ∧ 1𝑜 ⊆ 𝑦) → (((𝑦 ∈ N ∧
1𝑜 <N 𝑦) → 𝜒) → ((𝑦 ∈ N ∧
1𝑜 <N suc 𝑦) → 𝜃))) |
117 | 25, 30, 82, 86, 87, 116 | findsg 7093 |
. . . . . 6
⊢ (((𝐴 ∈ ω ∧
1𝑜 ∈ ω) ∧ 1𝑜 ⊆ 𝐴) → ((𝐴 ∈ N ∧
1𝑜 <N 𝐴) → 𝜏)) |
118 | 21, 117 | mpanl2 717 |
. . . . 5
⊢ ((𝐴 ∈ ω ∧
1𝑜 ⊆ 𝐴) → ((𝐴 ∈ N ∧
1𝑜 <N 𝐴) → 𝜏)) |
119 | 11, 20, 118 | syl2anc 693 |
. . . 4
⊢ (𝐴 ∈ N →
((𝐴 ∈ N
∧ 1𝑜 <N 𝐴) → 𝜏)) |
120 | 119 | expd 452 |
. . 3
⊢ (𝐴 ∈ N →
(𝐴 ∈ N
→ (1𝑜 <N 𝐴 → 𝜏))) |
121 | 120 | pm2.43i 52 |
. 2
⊢ (𝐴 ∈ N →
(1𝑜 <N 𝐴 → 𝜏)) |
122 | | nlt1pi 9728 |
. . . 4
⊢ ¬
𝐴
<N 1𝑜 |
123 | | ltsopi 9710 |
. . . . . 6
⊢
<N Or N |
124 | | sotric 5061 |
. . . . . 6
⊢ ((
<N Or N ∧ (𝐴 ∈ N ∧
1𝑜 ∈ N)) → (𝐴 <N
1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨
1𝑜 <N 𝐴))) |
125 | 123, 124 | mpan 706 |
. . . . 5
⊢ ((𝐴 ∈ N ∧
1𝑜 ∈ N) → (𝐴 <N
1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨
1𝑜 <N 𝐴))) |
126 | 1, 125 | mpan2 707 |
. . . 4
⊢ (𝐴 ∈ N →
(𝐴
<N 1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨
1𝑜 <N 𝐴))) |
127 | 122, 126 | mtbii 316 |
. . 3
⊢ (𝐴 ∈ N →
¬ ¬ (𝐴 =
1𝑜 ∨ 1𝑜 <N
𝐴)) |
128 | 127 | notnotrd 128 |
. 2
⊢ (𝐴 ∈ N →
(𝐴 = 1𝑜
∨ 1𝑜 <N 𝐴)) |
129 | 10, 121, 128 | mpjaod 396 |
1
⊢ (𝐴 ∈ N →
𝜏) |