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Theorem indpi 10897
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 8471 . . . . . 6 1o ∈ V
21eqvinc 3629 . . . . 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 3508 . . . 4 (1o = 𝐴𝜏)
87eqcoms 2732 . . 3 (𝐴 = 1o𝜏)
98a1i 11 . 2 (𝐴N → (𝐴 = 1o𝜏))
10 pinn 10868 . . . . 5 (𝐴N𝐴 ∈ ω)
11 elni2 10867 . . . . . 6 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
12 nnord 7856 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
13 ordsucss 7799 . . . . . . . . 9 (Ord 𝐴 → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
1412, 13syl 17 . . . . . . . 8 (𝐴 ∈ ω → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
15 df-1o 8461 . . . . . . . . 9 1o = suc ∅
1615sseq1i 4002 . . . . . . . 8 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
1714, 16imbitrrdi 251 . . . . . . 7 (𝐴 ∈ ω → (∅ ∈ 𝐴 → 1o𝐴))
1817imp 406 . . . . . 6 ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → 1o𝐴)
1911, 18sylbi 216 . . . . 5 (𝐴N → 1o𝐴)
20 1onn 8634 . . . . . 6 1o ∈ ω
21 eleq1 2813 . . . . . . . . 9 (𝑥 = 1o → (𝑥N ↔ 1oN))
22 breq2 5142 . . . . . . . . 9 (𝑥 = 1o → (1o <N 𝑥 ↔ 1o <N 1o))
2321, 22anbi12d 630 . . . . . . . 8 (𝑥 = 1o → ((𝑥N ∧ 1o <N 𝑥) ↔ (1oN ∧ 1o <N 1o)))
2423, 5imbi12d 344 . . . . . . 7 (𝑥 = 1o → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((1oN ∧ 1o <N 1o) → 𝜓)))
25 eleq1 2813 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥N𝑦N))
26 breq2 5142 . . . . . . . . 9 (𝑥 = 𝑦 → (1o <N 𝑥 ↔ 1o <N 𝑦))
2725, 26anbi12d 630 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥N ∧ 1o <N 𝑥) ↔ (𝑦N ∧ 1o <N 𝑦)))
28 indpi.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
2927, 28imbi12d 344 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1o <N 𝑦) → 𝜒)))
30 pinn 10868 . . . . . . . . . . . . . . 15 (𝑥N𝑥 ∈ ω)
31 eleq1 2813 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ suc 𝑦 ∈ ω))
32 peano2b 7865 . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦 ∈ ω))
36 1pi 10873 . . . . . . . . . . . . . . . 16 1oN
37 ltpiord 10877 . . . . . . . . . . . . . . . 16 ((1oN𝑥N) → (1o <N 𝑥 ↔ 1o𝑥))
3836, 37mpan 687 . . . . . . . . . . . . . . 15 (𝑥N → (1o <N 𝑥 ↔ 1o𝑥))
3938biimpa 476 . . . . . . . . . . . . . 14 ((𝑥N ∧ 1o <N 𝑥) → 1o𝑥)
40 eleq2 2814 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → (1o𝑥 ↔ 1o ∈ suc 𝑦))
41 elsuci 6421 . . . . . . . . . . . . . . . 16 (1o ∈ suc 𝑦 → (1o𝑦 ∨ 1o = 𝑦))
42 ne0i 4326 . . . . . . . . . . . . . . . . 17 (1o𝑦𝑦 ≠ ∅)
43 0lt1o 8499 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ 1o
44 eleq2 2814 . . . . . . . . . . . . . . . . . . 19 (1o = 𝑦 → (∅ ∈ 1o ↔ ∅ ∈ 𝑦))
4543, 44mpbii 232 . . . . . . . . . . . . . . . . . 18 (1o = 𝑦 → ∅ ∈ 𝑦)
4645ne0d 4327 . . . . . . . . . . . . . . . . 17 (1o = 𝑦𝑦 ≠ ∅)
4742, 46jaoi 854 . . . . . . . . . . . . . . . 16 ((1o𝑦 ∨ 1o = 𝑦) → 𝑦 ≠ ∅)
4841, 47syl 17 . . . . . . . . . . . . . . 15 (1o ∈ suc 𝑦𝑦 ≠ ∅)
4940, 48biimtrdi 252 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (1o𝑥𝑦 ≠ ∅))
5039, 49syl5 34 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦 ≠ ∅))
5135, 50jcad 512 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
52 elni 10866 . . . . . . . . . . . 12 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
5351, 52imbitrrdi 251 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦N))
54 simpr 484 . . . . . . . . . . . 12 ((𝑥N ∧ 1o <N 𝑥) → 1o <N 𝑥)
55 breq2 5142 . . . . . . . . . . . 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 512 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → (𝑦N ∧ 1o <N suc 𝑦)))
58 addclpi 10882 . . . . . . . . . . . . . . 15 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) ∈ N)
5936, 58mpan2 688 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1o) ∈ N)
60 addpiord 10874 . . . . . . . . . . . . . . . . . . 19 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) = (𝑦 +o 1o))
6136, 60mpan2 688 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +N 1o) = (𝑦 +o 1o))
62 pion 10869 . . . . . . . . . . . . . . . . . . 19 (𝑦N𝑦 ∈ On)
63 oa1suc 8526 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +o 1o) = suc 𝑦)
6561, 64eqtrd 2764 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1o) = suc 𝑦)
6665eqeq2d 2735 . . . . . . . . . . . . . . . 16 (𝑦N → (𝑥 = (𝑦 +N 1o) ↔ 𝑥 = suc 𝑦))
6766biimparc 479 . . . . . . . . . . . . . . 15 ((𝑥 = suc 𝑦𝑦N) → 𝑥 = (𝑦 +N 1o))
6867eleq1d 2810 . . . . . . . . . . . . . 14 ((𝑥 = suc 𝑦𝑦N) → (𝑥N ↔ (𝑦 +N 1o) ∈ N))
6959, 68imbitrrid 245 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑦N) → (𝑦N𝑥N))
7069ex 412 . . . . . . . . . . . 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 608 . . . . . . . . . 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 341 . . . . . . . 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 480 . . . . . . . . . 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 2813 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥N𝐴N))
83 breq2 5142 . . . . . . . . 9 (𝑥 = 𝐴 → (1o <N 𝑥 ↔ 1o <N 𝐴))
8482, 83anbi12d 630 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥N ∧ 1o <N 𝑥) ↔ (𝐴N ∧ 1o <N 𝐴)))
8584, 3imbi12d 344 . . . . . . 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 10877 . . . . . . . . . . . . . . 15 ((1oN𝑦N) → (1o <N 𝑦 ↔ 1o𝑦))
8836, 87mpan 687 . . . . . . . . . . . . . 14 (𝑦N → (1o <N 𝑦 ↔ 1o𝑦))
8988pm5.32i 574 . . . . . . . . . . . . 13 ((𝑦N ∧ 1o <N 𝑦) ↔ (𝑦N ∧ 1o𝑦))
9089simplbi2 500 . . . . . . . . . . . 12 (𝑦N → (1o𝑦 → (𝑦N ∧ 1o <N 𝑦)))
9190imim1d 82 . . . . . . . . . . 11 (𝑦N → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → (1o𝑦𝜒)))
92 ltrelpi 10879 . . . . . . . . . . . . . . 15 <N ⊆ (N × N)
9392brel 5731 . . . . . . . . . . . . . 14 (1o <N suc 𝑦 → (1oN ∧ suc 𝑦N))
94 ltpiord 10877 . . . . . . . . . . . . . 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 3629 . . . . . . . . . . . . . . 15 (1o = 𝑦 ↔ ∃𝑥(𝑥 = 1o𝑥 = 𝑦))
9897, 28, 6gencl 3508 . . . . . . . . . . . . . 14 (1o = 𝑦𝜒)
99 jao 957 . . . . . . . . . . . . . 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 410 . . . . . . . . 9 (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜒))
10515sseq1i 4002 . . . . . . . . . . 11 (1o𝑦 ↔ suc ∅ ⊆ 𝑦)
106 0ex 5297 . . . . . . . . . . . 12 ∅ ∈ V
107 sucssel 6449 . . . . . . . . . . . 12 (∅ ∈ V → (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦))
108106, 107ax-mp 5 . . . . . . . . . . 11 (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦)
109105, 108sylbi 216 . . . . . . . . . 10 (1o𝑦 → ∅ ∈ 𝑦)
110 elni2 10867 . . . . . . . . . . 11 (𝑦N ↔ (𝑦 ∈ ω ∧ ∅ ∈ 𝑦))
111 indpi.6 . . . . . . . . . . 11 (𝑦N → (𝜒𝜃))
112110, 111sylbir 234 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∅ ∈ 𝑦) → (𝜒𝜃))
113109, 112sylan2 592 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 1o𝑦) → (𝜒𝜃))
114104, 113syl9r 78 . . . . . . . 8 ((𝑦 ∈ ω ∧ 1o𝑦) → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
115114adantlr 712 . . . . . . 7 (((𝑦 ∈ ω ∧ 1o ∈ ω) ∧ 1o𝑦) → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
11624, 29, 81, 85, 86, 115findsg 7883 . . . . . 6 (((𝐴 ∈ ω ∧ 1o ∈ ω) ∧ 1o𝐴) → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
11720, 116mpanl2 698 . . . . 5 ((𝐴 ∈ ω ∧ 1o𝐴) → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
11810, 19, 117syl2anc 583 . . . 4 (𝐴N → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
119118expd 415 . . 3 (𝐴N → (𝐴N → (1o <N 𝐴𝜏)))
120119pm2.43i 52 . 2 (𝐴N → (1o <N 𝐴𝜏))
121 nlt1pi 10896 . . . 4 ¬ 𝐴 <N 1o
122 ltsopi 10878 . . . . . 6 <N Or N
123 sotric 5606 . . . . . 6 (( <N Or N ∧ (𝐴N ∧ 1oN)) → (𝐴 <N 1o ↔ ¬ (𝐴 = 1o ∨ 1o <N 𝐴)))
124122, 123mpan 687 . . . . 5 ((𝐴N ∧ 1oN) → (𝐴 <N 1o ↔ ¬ (𝐴 = 1o ∨ 1o <N 𝐴)))
12536, 124mpan2 688 . . . 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 857 1 (𝐴N𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844   = wceq 1533  wcel 2098  wne 2932  Vcvv 3466  wss 3940  c0 4314   class class class wbr 5138   Or wor 5577  Ord word 6353  Oncon0 6354  suc csuc 6356  (class class class)co 7401  ωcom 7848  1oc1o 8454   +o coa 8458  Ncnpi 10834   +N cpli 10835   <N clti 10837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-ni 10862  df-pli 10863  df-lti 10865
This theorem is referenced by:  prlem934  11023
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