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Theorem indpi 10902
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 1oex 8476 . . . . . 6 1o ∈ V
21eqvinc 3638 . . . . 5 (1o = 𝐴 ↔ ∃𝑥(𝑥 = 1o𝑥 = 𝐴))
3 indpi.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
4 indpi.5 . . . . . 6 𝜓
5 indpi.1 . . . . . 6 (𝑥 = 1o → (𝜑𝜓))
64, 5mpbiri 258 . . . . 5 (𝑥 = 1o𝜑)
72, 3, 6gencl 3516 . . . 4 (1o = 𝐴𝜏)
87eqcoms 2741 . . 3 (𝐴 = 1o𝜏)
98a1i 11 . 2 (𝐴N → (𝐴 = 1o𝜏))
10 pinn 10873 . . . . 5 (𝐴N𝐴 ∈ ω)
11 elni2 10872 . . . . . 6 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
12 nnord 7863 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
13 ordsucss 7806 . . . . . . . . 9 (Ord 𝐴 → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
1412, 13syl 17 . . . . . . . 8 (𝐴 ∈ ω → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
15 df-1o 8466 . . . . . . . . 9 1o = suc ∅
1615sseq1i 4011 . . . . . . . 8 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
1714, 16imbitrrdi 251 . . . . . . 7 (𝐴 ∈ ω → (∅ ∈ 𝐴 → 1o𝐴))
1817imp 408 . . . . . 6 ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → 1o𝐴)
1911, 18sylbi 216 . . . . 5 (𝐴N → 1o𝐴)
20 1onn 8639 . . . . . 6 1o ∈ ω
21 eleq1 2822 . . . . . . . . 9 (𝑥 = 1o → (𝑥N ↔ 1oN))
22 breq2 5153 . . . . . . . . 9 (𝑥 = 1o → (1o <N 𝑥 ↔ 1o <N 1o))
2321, 22anbi12d 632 . . . . . . . 8 (𝑥 = 1o → ((𝑥N ∧ 1o <N 𝑥) ↔ (1oN ∧ 1o <N 1o)))
2423, 5imbi12d 345 . . . . . . 7 (𝑥 = 1o → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((1oN ∧ 1o <N 1o) → 𝜓)))
25 eleq1 2822 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥N𝑦N))
26 breq2 5153 . . . . . . . . 9 (𝑥 = 𝑦 → (1o <N 𝑥 ↔ 1o <N 𝑦))
2725, 26anbi12d 632 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥N ∧ 1o <N 𝑥) ↔ (𝑦N ∧ 1o <N 𝑦)))
28 indpi.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
2927, 28imbi12d 345 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1o <N 𝑦) → 𝜒)))
30 pinn 10873 . . . . . . . . . . . . . . 15 (𝑥N𝑥 ∈ ω)
31 eleq1 2822 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ suc 𝑦 ∈ ω))
32 peano2b 7872 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
3331, 32bitr4di 289 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
3430, 33imbitrid 243 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝑥N𝑦 ∈ ω))
3534adantrd 493 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦 ∈ ω))
36 1pi 10878 . . . . . . . . . . . . . . . 16 1oN
37 ltpiord 10882 . . . . . . . . . . . . . . . 16 ((1oN𝑥N) → (1o <N 𝑥 ↔ 1o𝑥))
3836, 37mpan 689 . . . . . . . . . . . . . . 15 (𝑥N → (1o <N 𝑥 ↔ 1o𝑥))
3938biimpa 478 . . . . . . . . . . . . . 14 ((𝑥N ∧ 1o <N 𝑥) → 1o𝑥)
40 eleq2 2823 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → (1o𝑥 ↔ 1o ∈ suc 𝑦))
41 elsuci 6432 . . . . . . . . . . . . . . . 16 (1o ∈ suc 𝑦 → (1o𝑦 ∨ 1o = 𝑦))
42 ne0i 4335 . . . . . . . . . . . . . . . . 17 (1o𝑦𝑦 ≠ ∅)
43 0lt1o 8504 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ 1o
44 eleq2 2823 . . . . . . . . . . . . . . . . . . 19 (1o = 𝑦 → (∅ ∈ 1o ↔ ∅ ∈ 𝑦))
4543, 44mpbii 232 . . . . . . . . . . . . . . . . . 18 (1o = 𝑦 → ∅ ∈ 𝑦)
4645ne0d 4336 . . . . . . . . . . . . . . . . 17 (1o = 𝑦𝑦 ≠ ∅)
4742, 46jaoi 856 . . . . . . . . . . . . . . . 16 ((1o𝑦 ∨ 1o = 𝑦) → 𝑦 ≠ ∅)
4841, 47syl 17 . . . . . . . . . . . . . . 15 (1o ∈ suc 𝑦𝑦 ≠ ∅)
4940, 48syl6bi 253 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (1o𝑥𝑦 ≠ ∅))
5039, 49syl5 34 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦 ≠ ∅))
5135, 50jcad 514 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
52 elni 10871 . . . . . . . . . . . 12 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
5351, 52imbitrrdi 251 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦N))
54 simpr 486 . . . . . . . . . . . 12 ((𝑥N ∧ 1o <N 𝑥) → 1o <N 𝑥)
55 breq2 5153 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (1o <N 𝑥 ↔ 1o <N suc 𝑦))
5654, 55imbitrid 243 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 1o <N suc 𝑦))
5753, 56jcad 514 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → (𝑦N ∧ 1o <N suc 𝑦)))
58 addclpi 10887 . . . . . . . . . . . . . . 15 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) ∈ N)
5936, 58mpan2 690 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1o) ∈ N)
60 addpiord 10879 . . . . . . . . . . . . . . . . . . 19 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) = (𝑦 +o 1o))
6136, 60mpan2 690 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +N 1o) = (𝑦 +o 1o))
62 pion 10874 . . . . . . . . . . . . . . . . . . 19 (𝑦N𝑦 ∈ On)
63 oa1suc 8531 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +o 1o) = suc 𝑦)
6561, 64eqtrd 2773 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1o) = suc 𝑦)
6665eqeq2d 2744 . . . . . . . . . . . . . . . 16 (𝑦N → (𝑥 = (𝑦 +N 1o) ↔ 𝑥 = suc 𝑦))
6766biimparc 481 . . . . . . . . . . . . . . 15 ((𝑥 = suc 𝑦𝑦N) → 𝑥 = (𝑦 +N 1o))
6867eleq1d 2819 . . . . . . . . . . . . . 14 ((𝑥 = suc 𝑦𝑦N) → (𝑥N ↔ (𝑦 +N 1o) ∈ N))
6959, 68imbitrrid 245 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑦N) → (𝑦N𝑥N))
7069ex 414 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (𝑦N → (𝑦N𝑥N)))
7170pm2.43d 53 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝑦N𝑥N))
7255biimprd 247 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (1o <N suc 𝑦 → 1o <N 𝑥))
7371, 72anim12d 610 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝑦N ∧ 1o <N suc 𝑦) → (𝑥N ∧ 1o <N 𝑥)))
7457, 73impbid 211 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) ↔ (𝑦N ∧ 1o <N suc 𝑦)))
7574imbi1d 342 . . . . . . . 8 (𝑥 = suc 𝑦 → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1o <N suc 𝑦) → 𝜑)))
76 indpi.3 . . . . . . . . . . . 12 (𝑥 = (𝑦 +N 1o) → (𝜑𝜃))
7766, 76syl6bir 254 . . . . . . . . . . 11 (𝑦N → (𝑥 = suc 𝑦 → (𝜑𝜃)))
7877adantr 482 . . . . . . . . . 10 ((𝑦N ∧ 1o <N suc 𝑦) → (𝑥 = suc 𝑦 → (𝜑𝜃)))
7978com12 32 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝑦N ∧ 1o <N suc 𝑦) → (𝜑𝜃)))
8079pm5.74d 273 . . . . . . . 8 (𝑥 = suc 𝑦 → (((𝑦N ∧ 1o <N suc 𝑦) → 𝜑) ↔ ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
8175, 80bitrd 279 . . . . . . 7 (𝑥 = suc 𝑦 → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
82 eleq1 2822 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥N𝐴N))
83 breq2 5153 . . . . . . . . 9 (𝑥 = 𝐴 → (1o <N 𝑥 ↔ 1o <N 𝐴))
8482, 83anbi12d 632 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥N ∧ 1o <N 𝑥) ↔ (𝐴N ∧ 1o <N 𝐴)))
8584, 3imbi12d 345 . . . . . . 7 (𝑥 = 𝐴 → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((𝐴N ∧ 1o <N 𝐴) → 𝜏)))
8642a1i 12 . . . . . . 7 (1o ∈ ω → ((1oN ∧ 1o <N 1o) → 𝜓))
87 ltpiord 10882 . . . . . . . . . . . . . . 15 ((1oN𝑦N) → (1o <N 𝑦 ↔ 1o𝑦))
8836, 87mpan 689 . . . . . . . . . . . . . 14 (𝑦N → (1o <N 𝑦 ↔ 1o𝑦))
8988pm5.32i 576 . . . . . . . . . . . . 13 ((𝑦N ∧ 1o <N 𝑦) ↔ (𝑦N ∧ 1o𝑦))
9089simplbi2 502 . . . . . . . . . . . 12 (𝑦N → (1o𝑦 → (𝑦N ∧ 1o <N 𝑦)))
9190imim1d 82 . . . . . . . . . . 11 (𝑦N → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → (1o𝑦𝜒)))
92 ltrelpi 10884 . . . . . . . . . . . . . . 15 <N ⊆ (N × N)
9392brel 5742 . . . . . . . . . . . . . 14 (1o <N suc 𝑦 → (1oN ∧ suc 𝑦N))
94 ltpiord 10882 . . . . . . . . . . . . . 14 ((1oN ∧ suc 𝑦N) → (1o <N suc 𝑦 ↔ 1o ∈ suc 𝑦))
9593, 94syl 17 . . . . . . . . . . . . 13 (1o <N suc 𝑦 → (1o <N suc 𝑦 ↔ 1o ∈ suc 𝑦))
9695ibi 267 . . . . . . . . . . . 12 (1o <N suc 𝑦 → 1o ∈ suc 𝑦)
971eqvinc 3638 . . . . . . . . . . . . . . 15 (1o = 𝑦 ↔ ∃𝑥(𝑥 = 1o𝑥 = 𝑦))
9897, 28, 6gencl 3516 . . . . . . . . . . . . . 14 (1o = 𝑦𝜒)
99 jao 960 . . . . . . . . . . . . . 14 ((1o𝑦𝜒) → ((1o = 𝑦𝜒) → ((1o𝑦 ∨ 1o = 𝑦) → 𝜒)))
10098, 99mpi 20 . . . . . . . . . . . . 13 ((1o𝑦𝜒) → ((1o𝑦 ∨ 1o = 𝑦) → 𝜒))
10141, 100syl5 34 . . . . . . . . . . . 12 ((1o𝑦𝜒) → (1o ∈ suc 𝑦𝜒))
10296, 101syl5 34 . . . . . . . . . . 11 ((1o𝑦𝜒) → (1o <N suc 𝑦𝜒))
10391, 102syl6com 37 . . . . . . . . . 10 (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → (𝑦N → (1o <N suc 𝑦𝜒)))
104103impd 412 . . . . . . . . 9 (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜒))
10515sseq1i 4011 . . . . . . . . . . 11 (1o𝑦 ↔ suc ∅ ⊆ 𝑦)
106 0ex 5308 . . . . . . . . . . . 12 ∅ ∈ V
107 sucssel 6460 . . . . . . . . . . . 12 (∅ ∈ V → (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦))
108106, 107ax-mp 5 . . . . . . . . . . 11 (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦)
109105, 108sylbi 216 . . . . . . . . . 10 (1o𝑦 → ∅ ∈ 𝑦)
110 elni2 10872 . . . . . . . . . . 11 (𝑦N ↔ (𝑦 ∈ ω ∧ ∅ ∈ 𝑦))
111 indpi.6 . . . . . . . . . . 11 (𝑦N → (𝜒𝜃))
112110, 111sylbir 234 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∅ ∈ 𝑦) → (𝜒𝜃))
113109, 112sylan2 594 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 1o𝑦) → (𝜒𝜃))
114104, 113syl9r 78 . . . . . . . 8 ((𝑦 ∈ ω ∧ 1o𝑦) → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
115114adantlr 714 . . . . . . 7 (((𝑦 ∈ ω ∧ 1o ∈ ω) ∧ 1o𝑦) → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
11624, 29, 81, 85, 86, 115findsg 7890 . . . . . 6 (((𝐴 ∈ ω ∧ 1o ∈ ω) ∧ 1o𝐴) → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
11720, 116mpanl2 700 . . . . 5 ((𝐴 ∈ ω ∧ 1o𝐴) → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
11810, 19, 117syl2anc 585 . . . 4 (𝐴N → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
119118expd 417 . . 3 (𝐴N → (𝐴N → (1o <N 𝐴𝜏)))
120119pm2.43i 52 . 2 (𝐴N → (1o <N 𝐴𝜏))
121 nlt1pi 10901 . . . 4 ¬ 𝐴 <N 1o
122 ltsopi 10883 . . . . . 6 <N Or N
123 sotric 5617 . . . . . 6 (( <N Or N ∧ (𝐴N ∧ 1oN)) → (𝐴 <N 1o ↔ ¬ (𝐴 = 1o ∨ 1o <N 𝐴)))
124122, 123mpan 689 . . . . 5 ((𝐴N ∧ 1oN) → (𝐴 <N 1o ↔ ¬ (𝐴 = 1o ∨ 1o <N 𝐴)))
12536, 124mpan2 690 . . . 4 (𝐴N → (𝐴 <N 1o ↔ ¬ (𝐴 = 1o ∨ 1o <N 𝐴)))
126121, 125mtbii 326 . . 3 (𝐴N → ¬ ¬ (𝐴 = 1o ∨ 1o <N 𝐴))
127126notnotrd 133 . 2 (𝐴N → (𝐴 = 1o ∨ 1o <N 𝐴))
1289, 120, 127mpjaod 859 1 (𝐴N𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  wss 3949  c0 4323   class class class wbr 5149   Or wor 5588  Ord word 6364  Oncon0 6365  suc csuc 6367  (class class class)co 7409  ωcom 7855  1oc1o 8459   +o coa 8463  Ncnpi 10839   +N cpli 10840   <N clti 10842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-ni 10867  df-pli 10868  df-lti 10870
This theorem is referenced by:  prlem934  11028
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