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Theorem indpi 9767
Description: Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
Hypotheses
Ref Expression
indpi.1 (𝑥 = 1𝑜 → (𝜑𝜓))
indpi.2 (𝑥 = 𝑦 → (𝜑𝜒))
indpi.3 (𝑥 = (𝑦 +N 1𝑜) → (𝜑𝜃))
indpi.4 (𝑥 = 𝐴 → (𝜑𝜏))
indpi.5 𝜓
indpi.6 (𝑦N → (𝜒𝜃))
Assertion
Ref Expression
indpi (𝐴N𝜏)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem indpi
StepHypRef Expression
1 1pi 9743 . . . . . . 7 1𝑜N
21elexi 3244 . . . . . 6 1𝑜 ∈ V
32eqvinc 3361 . . . . 5 (1𝑜 = 𝐴 ↔ ∃𝑥(𝑥 = 1𝑜𝑥 = 𝐴))
4 indpi.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
5 indpi.5 . . . . . 6 𝜓
6 indpi.1 . . . . . 6 (𝑥 = 1𝑜 → (𝜑𝜓))
75, 6mpbiri 248 . . . . 5 (𝑥 = 1𝑜𝜑)
83, 4, 7gencl 3266 . . . 4 (1𝑜 = 𝐴𝜏)
98eqcoms 2659 . . 3 (𝐴 = 1𝑜𝜏)
109a1i 11 . 2 (𝐴N → (𝐴 = 1𝑜𝜏))
11 pinn 9738 . . . . 5 (𝐴N𝐴 ∈ ω)
12 elni2 9737 . . . . . 6 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
13 nnord 7115 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
14 ordsucss 7060 . . . . . . . . 9 (Ord 𝐴 → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
1513, 14syl 17 . . . . . . . 8 (𝐴 ∈ ω → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
16 df-1o 7605 . . . . . . . . 9 1𝑜 = suc ∅
1716sseq1i 3662 . . . . . . . 8 (1𝑜𝐴 ↔ suc ∅ ⊆ 𝐴)
1815, 17syl6ibr 242 . . . . . . 7 (𝐴 ∈ ω → (∅ ∈ 𝐴 → 1𝑜𝐴))
1918imp 444 . . . . . 6 ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → 1𝑜𝐴)
2012, 19sylbi 207 . . . . 5 (𝐴N → 1𝑜𝐴)
21 1onn 7764 . . . . . 6 1𝑜 ∈ ω
22 eleq1 2718 . . . . . . . . 9 (𝑥 = 1𝑜 → (𝑥N ↔ 1𝑜N))
23 breq2 4689 . . . . . . . . 9 (𝑥 = 1𝑜 → (1𝑜 <N 𝑥 ↔ 1𝑜 <N 1𝑜))
2422, 23anbi12d 747 . . . . . . . 8 (𝑥 = 1𝑜 → ((𝑥N ∧ 1𝑜 <N 𝑥) ↔ (1𝑜N ∧ 1𝑜 <N 1𝑜)))
2524, 6imbi12d 333 . . . . . . 7 (𝑥 = 1𝑜 → (((𝑥N ∧ 1𝑜 <N 𝑥) → 𝜑) ↔ ((1𝑜N ∧ 1𝑜 <N 1𝑜) → 𝜓)))
26 eleq1 2718 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥N𝑦N))
27 breq2 4689 . . . . . . . . 9 (𝑥 = 𝑦 → (1𝑜 <N 𝑥 ↔ 1𝑜 <N 𝑦))
2826, 27anbi12d 747 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) ↔ (𝑦N ∧ 1𝑜 <N 𝑦)))
29 indpi.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
3028, 29imbi12d 333 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥N ∧ 1𝑜 <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒)))
31 pinn 9738 . . . . . . . . . . . . . . 15 (𝑥N𝑥 ∈ ω)
32 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ suc 𝑦 ∈ ω))
33 peano2b 7123 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
3432, 33syl6bbr 278 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
3531, 34syl5ib 234 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝑥N𝑦 ∈ ω))
3635adantrd 483 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → 𝑦 ∈ ω))
37 ltpiord 9747 . . . . . . . . . . . . . . . 16 ((1𝑜N𝑥N) → (1𝑜 <N 𝑥 ↔ 1𝑜𝑥))
381, 37mpan 706 . . . . . . . . . . . . . . 15 (𝑥N → (1𝑜 <N 𝑥 ↔ 1𝑜𝑥))
3938biimpa 500 . . . . . . . . . . . . . 14 ((𝑥N ∧ 1𝑜 <N 𝑥) → 1𝑜𝑥)
40 eleq2 2719 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → (1𝑜𝑥 ↔ 1𝑜 ∈ suc 𝑦))
41 elsuci 5829 . . . . . . . . . . . . . . . 16 (1𝑜 ∈ suc 𝑦 → (1𝑜𝑦 ∨ 1𝑜 = 𝑦))
42 ne0i 3954 . . . . . . . . . . . . . . . . 17 (1𝑜𝑦𝑦 ≠ ∅)
43 0lt1o 7629 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ 1𝑜
44 eleq2 2719 . . . . . . . . . . . . . . . . . . 19 (1𝑜 = 𝑦 → (∅ ∈ 1𝑜 ↔ ∅ ∈ 𝑦))
4543, 44mpbii 223 . . . . . . . . . . . . . . . . . 18 (1𝑜 = 𝑦 → ∅ ∈ 𝑦)
46 ne0i 3954 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 𝑦𝑦 ≠ ∅)
4745, 46syl 17 . . . . . . . . . . . . . . . . 17 (1𝑜 = 𝑦𝑦 ≠ ∅)
4842, 47jaoi 393 . . . . . . . . . . . . . . . 16 ((1𝑜𝑦 ∨ 1𝑜 = 𝑦) → 𝑦 ≠ ∅)
4941, 48syl 17 . . . . . . . . . . . . . . 15 (1𝑜 ∈ suc 𝑦𝑦 ≠ ∅)
5040, 49syl6bi 243 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (1𝑜𝑥𝑦 ≠ ∅))
5139, 50syl5 34 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → 𝑦 ≠ ∅))
5236, 51jcad 554 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
53 elni 9736 . . . . . . . . . . . 12 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
5452, 53syl6ibr 242 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → 𝑦N))
55 simpr 476 . . . . . . . . . . . 12 ((𝑥N ∧ 1𝑜 <N 𝑥) → 1𝑜 <N 𝑥)
56 breq2 4689 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (1𝑜 <N 𝑥 ↔ 1𝑜 <N suc 𝑦))
5755, 56syl5ib 234 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → 1𝑜 <N suc 𝑦))
5854, 57jcad 554 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → (𝑦N ∧ 1𝑜 <N suc 𝑦)))
59 addclpi 9752 . . . . . . . . . . . . . . 15 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) ∈ N)
601, 59mpan2 707 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1𝑜) ∈ N)
61 addpiord 9744 . . . . . . . . . . . . . . . . . . 19 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
621, 61mpan2 707 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
63 pion 9739 . . . . . . . . . . . . . . . . . . 19 (𝑦N𝑦 ∈ On)
64 oa1suc 7656 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
6563, 64syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +𝑜 1𝑜) = suc 𝑦)
6662, 65eqtrd 2685 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1𝑜) = suc 𝑦)
6766eqeq2d 2661 . . . . . . . . . . . . . . . 16 (𝑦N → (𝑥 = (𝑦 +N 1𝑜) ↔ 𝑥 = suc 𝑦))
6867biimparc 503 . . . . . . . . . . . . . . 15 ((𝑥 = suc 𝑦𝑦N) → 𝑥 = (𝑦 +N 1𝑜))
6968eleq1d 2715 . . . . . . . . . . . . . 14 ((𝑥 = suc 𝑦𝑦N) → (𝑥N ↔ (𝑦 +N 1𝑜) ∈ N))
7060, 69syl5ibr 236 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑦N) → (𝑦N𝑥N))
7170ex 449 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (𝑦N → (𝑦N𝑥N)))
7271pm2.43d 53 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝑦N𝑥N))
7356biimprd 238 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (1𝑜 <N suc 𝑦 → 1𝑜 <N 𝑥))
7472, 73anim12d 585 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝑦N ∧ 1𝑜 <N suc 𝑦) → (𝑥N ∧ 1𝑜 <N 𝑥)))
7558, 74impbid 202 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) ↔ (𝑦N ∧ 1𝑜 <N suc 𝑦)))
7675imbi1d 330 . . . . . . . 8 (𝑥 = suc 𝑦 → (((𝑥N ∧ 1𝑜 <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜑)))
77 indpi.3 . . . . . . . . . . . 12 (𝑥 = (𝑦 +N 1𝑜) → (𝜑𝜃))
7867, 77syl6bir 244 . . . . . . . . . . 11 (𝑦N → (𝑥 = suc 𝑦 → (𝜑𝜃)))
7978adantr 480 . . . . . . . . . 10 ((𝑦N ∧ 1𝑜 <N suc 𝑦) → (𝑥 = suc 𝑦 → (𝜑𝜃)))
8079com12 32 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝑦N ∧ 1𝑜 <N suc 𝑦) → (𝜑𝜃)))
8180pm5.74d 262 . . . . . . . 8 (𝑥 = suc 𝑦 → (((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜑) ↔ ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜃)))
8276, 81bitrd 268 . . . . . . 7 (𝑥 = suc 𝑦 → (((𝑥N ∧ 1𝑜 <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜃)))
83 eleq1 2718 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥N𝐴N))
84 breq2 4689 . . . . . . . . 9 (𝑥 = 𝐴 → (1𝑜 <N 𝑥 ↔ 1𝑜 <N 𝐴))
8583, 84anbi12d 747 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥N ∧ 1𝑜 <N 𝑥) ↔ (𝐴N ∧ 1𝑜 <N 𝐴)))
8685, 4imbi12d 333 . . . . . . 7 (𝑥 = 𝐴 → (((𝑥N ∧ 1𝑜 <N 𝑥) → 𝜑) ↔ ((𝐴N ∧ 1𝑜 <N 𝐴) → 𝜏)))
8752a1i 12 . . . . . . 7 (1𝑜 ∈ ω → ((1𝑜N ∧ 1𝑜 <N 1𝑜) → 𝜓))
88 ltpiord 9747 . . . . . . . . . . . . . . 15 ((1𝑜N𝑦N) → (1𝑜 <N 𝑦 ↔ 1𝑜𝑦))
891, 88mpan 706 . . . . . . . . . . . . . 14 (𝑦N → (1𝑜 <N 𝑦 ↔ 1𝑜𝑦))
9089pm5.32i 670 . . . . . . . . . . . . 13 ((𝑦N ∧ 1𝑜 <N 𝑦) ↔ (𝑦N ∧ 1𝑜𝑦))
9190simplbi2 654 . . . . . . . . . . . 12 (𝑦N → (1𝑜𝑦 → (𝑦N ∧ 1𝑜 <N 𝑦)))
9291imim1d 82 . . . . . . . . . . 11 (𝑦N → (((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒) → (1𝑜𝑦𝜒)))
93 ltrelpi 9749 . . . . . . . . . . . . . . 15 <N ⊆ (N × N)
9493brel 5202 . . . . . . . . . . . . . 14 (1𝑜 <N suc 𝑦 → (1𝑜N ∧ suc 𝑦N))
95 ltpiord 9747 . . . . . . . . . . . . . 14 ((1𝑜N ∧ suc 𝑦N) → (1𝑜 <N suc 𝑦 ↔ 1𝑜 ∈ suc 𝑦))
9694, 95syl 17 . . . . . . . . . . . . 13 (1𝑜 <N suc 𝑦 → (1𝑜 <N suc 𝑦 ↔ 1𝑜 ∈ suc 𝑦))
9796ibi 256 . . . . . . . . . . . 12 (1𝑜 <N suc 𝑦 → 1𝑜 ∈ suc 𝑦)
982eqvinc 3361 . . . . . . . . . . . . . . 15 (1𝑜 = 𝑦 ↔ ∃𝑥(𝑥 = 1𝑜𝑥 = 𝑦))
9998, 29, 7gencl 3266 . . . . . . . . . . . . . 14 (1𝑜 = 𝑦𝜒)
100 jao 533 . . . . . . . . . . . . . 14 ((1𝑜𝑦𝜒) → ((1𝑜 = 𝑦𝜒) → ((1𝑜𝑦 ∨ 1𝑜 = 𝑦) → 𝜒)))
10199, 100mpi 20 . . . . . . . . . . . . 13 ((1𝑜𝑦𝜒) → ((1𝑜𝑦 ∨ 1𝑜 = 𝑦) → 𝜒))
10241, 101syl5 34 . . . . . . . . . . . 12 ((1𝑜𝑦𝜒) → (1𝑜 ∈ suc 𝑦𝜒))
10397, 102syl5 34 . . . . . . . . . . 11 ((1𝑜𝑦𝜒) → (1𝑜 <N suc 𝑦𝜒))
10492, 103syl6com 37 . . . . . . . . . 10 (((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒) → (𝑦N → (1𝑜 <N suc 𝑦𝜒)))
105104impd 446 . . . . . . . . 9 (((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒) → ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜒))
10616sseq1i 3662 . . . . . . . . . . 11 (1𝑜𝑦 ↔ suc ∅ ⊆ 𝑦)
107 0ex 4823 . . . . . . . . . . . 12 ∅ ∈ V
108 sucssel 5857 . . . . . . . . . . . 12 (∅ ∈ V → (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦))
109107, 108ax-mp 5 . . . . . . . . . . 11 (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦)
110106, 109sylbi 207 . . . . . . . . . 10 (1𝑜𝑦 → ∅ ∈ 𝑦)
111 elni2 9737 . . . . . . . . . . 11 (𝑦N ↔ (𝑦 ∈ ω ∧ ∅ ∈ 𝑦))
112 indpi.6 . . . . . . . . . . 11 (𝑦N → (𝜒𝜃))
113111, 112sylbir 225 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∅ ∈ 𝑦) → (𝜒𝜃))
114110, 113sylan2 490 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 1𝑜𝑦) → (𝜒𝜃))
115105, 114syl9r 78 . . . . . . . 8 ((𝑦 ∈ ω ∧ 1𝑜𝑦) → (((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒) → ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜃)))
116115adantlr 751 . . . . . . 7 (((𝑦 ∈ ω ∧ 1𝑜 ∈ ω) ∧ 1𝑜𝑦) → (((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒) → ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜃)))
11725, 30, 82, 86, 87, 116findsg 7135 . . . . . 6 (((𝐴 ∈ ω ∧ 1𝑜 ∈ ω) ∧ 1𝑜𝐴) → ((𝐴N ∧ 1𝑜 <N 𝐴) → 𝜏))
11821, 117mpanl2 717 . . . . 5 ((𝐴 ∈ ω ∧ 1𝑜𝐴) → ((𝐴N ∧ 1𝑜 <N 𝐴) → 𝜏))
11911, 20, 118syl2anc 694 . . . 4 (𝐴N → ((𝐴N ∧ 1𝑜 <N 𝐴) → 𝜏))
120119expd 451 . . 3 (𝐴N → (𝐴N → (1𝑜 <N 𝐴𝜏)))
121120pm2.43i 52 . 2 (𝐴N → (1𝑜 <N 𝐴𝜏))
122 nlt1pi 9766 . . . 4 ¬ 𝐴 <N 1𝑜
123 ltsopi 9748 . . . . . 6 <N Or N
124 sotric 5090 . . . . . 6 (( <N Or N ∧ (𝐴N ∧ 1𝑜N)) → (𝐴 <N 1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴)))
125123, 124mpan 706 . . . . 5 ((𝐴N ∧ 1𝑜N) → (𝐴 <N 1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴)))
1261, 125mpan2 707 . . . 4 (𝐴N → (𝐴 <N 1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴)))
127122, 126mtbii 315 . . 3 (𝐴N → ¬ ¬ (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴))
128127notnotrd 128 . 2 (𝐴N → (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴))
12910, 121, 128mpjaod 395 1 (𝐴N𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  wss 3607  c0 3948   class class class wbr 4685   Or wor 5063  Ord word 5760  Oncon0 5761  suc csuc 5763  (class class class)co 6690  ωcom 7107  1𝑜c1o 7598   +𝑜 coa 7602  Ncnpi 9704   +N cpli 9705   <N clti 9707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-ni 9732  df-pli 9733  df-lti 9735
This theorem is referenced by:  prlem934  9893
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