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Theorem indpi 9681
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 9657 . . . . . . 7 1𝑜N
21elexi 3202 . . . . . 6 1𝑜 ∈ V
32eqvinc 3317 . . . . 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 3224 . . . 4 (1𝑜 = 𝐴𝜏)
98eqcoms 2629 . . 3 (𝐴 = 1𝑜𝜏)
109a1i 11 . 2 (𝐴N → (𝐴 = 1𝑜𝜏))
11 pinn 9652 . . . . 5 (𝐴N𝐴 ∈ ω)
12 elni2 9651 . . . . . 6 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
13 nnord 7027 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
14 ordsucss 6972 . . . . . . . . 9 (Ord 𝐴 → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
1513, 14syl 17 . . . . . . . 8 (𝐴 ∈ ω → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
16 df-1o 7512 . . . . . . . . 9 1𝑜 = suc ∅
1716sseq1i 3613 . . . . . . . 8 (1𝑜𝐴 ↔ suc ∅ ⊆ 𝐴)
1815, 17syl6ibr 242 . . . . . . 7 (𝐴 ∈ ω → (∅ ∈ 𝐴 → 1𝑜𝐴))
1918imp 445 . . . . . 6 ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → 1𝑜𝐴)
2012, 19sylbi 207 . . . . 5 (𝐴N → 1𝑜𝐴)
21 1onn 7671 . . . . . 6 1𝑜 ∈ ω
22 eleq1 2686 . . . . . . . . 9 (𝑥 = 1𝑜 → (𝑥N ↔ 1𝑜N))
23 breq2 4622 . . . . . . . . 9 (𝑥 = 1𝑜 → (1𝑜 <N 𝑥 ↔ 1𝑜 <N 1𝑜))
2422, 23anbi12d 746 . . . . . . . 8 (𝑥 = 1𝑜 → ((𝑥N ∧ 1𝑜 <N 𝑥) ↔ (1𝑜N ∧ 1𝑜 <N 1𝑜)))
2524, 6imbi12d 334 . . . . . . 7 (𝑥 = 1𝑜 → (((𝑥N ∧ 1𝑜 <N 𝑥) → 𝜑) ↔ ((1𝑜N ∧ 1𝑜 <N 1𝑜) → 𝜓)))
26 eleq1 2686 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥N𝑦N))
27 breq2 4622 . . . . . . . . 9 (𝑥 = 𝑦 → (1𝑜 <N 𝑥 ↔ 1𝑜 <N 𝑦))
2826, 27anbi12d 746 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) ↔ (𝑦N ∧ 1𝑜 <N 𝑦)))
29 indpi.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
3028, 29imbi12d 334 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥N ∧ 1𝑜 <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒)))
31 pinn 9652 . . . . . . . . . . . . . . 15 (𝑥N𝑥 ∈ ω)
32 eleq1 2686 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ suc 𝑦 ∈ ω))
33 peano2b 7035 . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → 𝑦 ∈ ω))
37 ltpiord 9661 . . . . . . . . . . . . . . . 16 ((1𝑜N𝑥N) → (1𝑜 <N 𝑥 ↔ 1𝑜𝑥))
381, 37mpan 705 . . . . . . . . . . . . . . 15 (𝑥N → (1𝑜 <N 𝑥 ↔ 1𝑜𝑥))
3938biimpa 501 . . . . . . . . . . . . . 14 ((𝑥N ∧ 1𝑜 <N 𝑥) → 1𝑜𝑥)
40 eleq2 2687 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → (1𝑜𝑥 ↔ 1𝑜 ∈ suc 𝑦))
41 elsuci 5755 . . . . . . . . . . . . . . . 16 (1𝑜 ∈ suc 𝑦 → (1𝑜𝑦 ∨ 1𝑜 = 𝑦))
42 ne0i 3902 . . . . . . . . . . . . . . . . 17 (1𝑜𝑦𝑦 ≠ ∅)
43 0lt1o 7536 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ 1𝑜
44 eleq2 2687 . . . . . . . . . . . . . . . . . . 19 (1𝑜 = 𝑦 → (∅ ∈ 1𝑜 ↔ ∅ ∈ 𝑦))
4543, 44mpbii 223 . . . . . . . . . . . . . . . . . 18 (1𝑜 = 𝑦 → ∅ ∈ 𝑦)
46 ne0i 3902 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 𝑦𝑦 ≠ ∅)
4745, 46syl 17 . . . . . . . . . . . . . . . . 17 (1𝑜 = 𝑦𝑦 ≠ ∅)
4842, 47jaoi 394 . . . . . . . . . . . . . . . 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 555 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
53 elni 9650 . . . . . . . . . . . 12 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
5452, 53syl6ibr 242 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → 𝑦N))
55 simpr 477 . . . . . . . . . . . 12 ((𝑥N ∧ 1𝑜 <N 𝑥) → 1𝑜 <N 𝑥)
56 breq2 4622 . . . . . . . . . . . 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 555 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1𝑜 <N 𝑥) → (𝑦N ∧ 1𝑜 <N suc 𝑦)))
59 addclpi 9666 . . . . . . . . . . . . . . 15 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) ∈ N)
601, 59mpan2 706 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1𝑜) ∈ N)
61 addpiord 9658 . . . . . . . . . . . . . . . . . . 19 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
621, 61mpan2 706 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
63 pion 9653 . . . . . . . . . . . . . . . . . . 19 (𝑦N𝑦 ∈ On)
64 oa1suc 7563 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
6563, 64syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +𝑜 1𝑜) = suc 𝑦)
6662, 65eqtrd 2655 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1𝑜) = suc 𝑦)
6766eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑦N → (𝑥 = (𝑦 +N 1𝑜) ↔ 𝑥 = suc 𝑦))
6867biimparc 504 . . . . . . . . . . . . . . 15 ((𝑥 = suc 𝑦𝑦N) → 𝑥 = (𝑦 +N 1𝑜))
6968eleq1d 2683 . . . . . . . . . . . . . 14 ((𝑥 = suc 𝑦𝑦N) → (𝑥N ↔ (𝑦 +N 1𝑜) ∈ N))
7060, 69syl5ibr 236 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑦N) → (𝑦N𝑥N))
7170ex 450 . . . . . . . . . . . 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 331 . . . . . . . 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 481 . . . . . . . . . 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 2686 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥N𝐴N))
84 breq2 4622 . . . . . . . . 9 (𝑥 = 𝐴 → (1𝑜 <N 𝑥 ↔ 1𝑜 <N 𝐴))
8583, 84anbi12d 746 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥N ∧ 1𝑜 <N 𝑥) ↔ (𝐴N ∧ 1𝑜 <N 𝐴)))
8685, 4imbi12d 334 . . . . . . 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 9661 . . . . . . . . . . . . . . 15 ((1𝑜N𝑦N) → (1𝑜 <N 𝑦 ↔ 1𝑜𝑦))
891, 88mpan 705 . . . . . . . . . . . . . 14 (𝑦N → (1𝑜 <N 𝑦 ↔ 1𝑜𝑦))
9089pm5.32i 668 . . . . . . . . . . . . 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 9663 . . . . . . . . . . . . . . 15 <N ⊆ (N × N)
9493brel 5133 . . . . . . . . . . . . . 14 (1𝑜 <N suc 𝑦 → (1𝑜N ∧ suc 𝑦N))
95 ltpiord 9661 . . . . . . . . . . . . . 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 3317 . . . . . . . . . . . . . . 15 (1𝑜 = 𝑦 ↔ ∃𝑥(𝑥 = 1𝑜𝑥 = 𝑦))
9998, 29, 7gencl 3224 . . . . . . . . . . . . . 14 (1𝑜 = 𝑦𝜒)
100 jao 534 . . . . . . . . . . . . . 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 447 . . . . . . . . 9 (((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒) → ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜒))
10616sseq1i 3613 . . . . . . . . . . 11 (1𝑜𝑦 ↔ suc ∅ ⊆ 𝑦)
107 0ex 4755 . . . . . . . . . . . 12 ∅ ∈ V
108 sucssel 5783 . . . . . . . . . . . 12 (∅ ∈ V → (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦))
109107, 108ax-mp 5 . . . . . . . . . . 11 (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦)
110106, 109sylbi 207 . . . . . . . . . 10 (1𝑜𝑦 → ∅ ∈ 𝑦)
111 elni2 9651 . . . . . . . . . . 11 (𝑦N ↔ (𝑦 ∈ ω ∧ ∅ ∈ 𝑦))
112 indpi.6 . . . . . . . . . . 11 (𝑦N → (𝜒𝜃))
113111, 112sylbir 225 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∅ ∈ 𝑦) → (𝜒𝜃))
114110, 113sylan2 491 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 1𝑜𝑦) → (𝜒𝜃))
115105, 114syl9r 78 . . . . . . . 8 ((𝑦 ∈ ω ∧ 1𝑜𝑦) → (((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒) → ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜃)))
116115adantlr 750 . . . . . . 7 (((𝑦 ∈ ω ∧ 1𝑜 ∈ ω) ∧ 1𝑜𝑦) → (((𝑦N ∧ 1𝑜 <N 𝑦) → 𝜒) → ((𝑦N ∧ 1𝑜 <N suc 𝑦) → 𝜃)))
11725, 30, 82, 86, 87, 116findsg 7047 . . . . . 6 (((𝐴 ∈ ω ∧ 1𝑜 ∈ ω) ∧ 1𝑜𝐴) → ((𝐴N ∧ 1𝑜 <N 𝐴) → 𝜏))
11821, 117mpanl2 716 . . . . 5 ((𝐴 ∈ ω ∧ 1𝑜𝐴) → ((𝐴N ∧ 1𝑜 <N 𝐴) → 𝜏))
11911, 20, 118syl2anc 692 . . . 4 (𝐴N → ((𝐴N ∧ 1𝑜 <N 𝐴) → 𝜏))
120119expd 452 . . 3 (𝐴N → (𝐴N → (1𝑜 <N 𝐴𝜏)))
121120pm2.43i 52 . 2 (𝐴N → (1𝑜 <N 𝐴𝜏))
122 nlt1pi 9680 . . . 4 ¬ 𝐴 <N 1𝑜
123 ltsopi 9662 . . . . . 6 <N Or N
124 sotric 5026 . . . . . 6 (( <N Or N ∧ (𝐴N ∧ 1𝑜N)) → (𝐴 <N 1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴)))
125123, 124mpan 705 . . . . 5 ((𝐴N ∧ 1𝑜N) → (𝐴 <N 1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴)))
1261, 125mpan2 706 . . . 4 (𝐴N → (𝐴 <N 1𝑜 ↔ ¬ (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴)))
127122, 126mtbii 316 . . 3 (𝐴N → ¬ ¬ (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴))
128127notnotrd 128 . 2 (𝐴N → (𝐴 = 1𝑜 ∨ 1𝑜 <N 𝐴))
12910, 121, 128mpjaod 396 1 (𝐴N𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3189  wss 3559  c0 3896   class class class wbr 4618   Or wor 4999  Ord word 5686  Oncon0 5687  suc csuc 5689  (class class class)co 6610  ωcom 7019  1𝑜c1o 7505   +𝑜 coa 7509  Ncnpi 9618   +N cpli 9619   <N clti 9621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-ni 9646  df-pli 9647  df-lti 9649
This theorem is referenced by:  prlem934  9807
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