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Theorem indpi 10360
 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 8121 . . . . . 6 1o ∈ V
21eqvinc 3561 . . . . 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 261 . . . . 5 (𝑥 = 1o𝜑)
72, 3, 6gencl 3451 . . . 4 (1o = 𝐴𝜏)
87eqcoms 2767 . . 3 (𝐴 = 1o𝜏)
98a1i 11 . 2 (𝐴N → (𝐴 = 1o𝜏))
10 pinn 10331 . . . . 5 (𝐴N𝐴 ∈ ω)
11 elni2 10330 . . . . . 6 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
12 nnord 7588 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
13 ordsucss 7533 . . . . . . . . 9 (Ord 𝐴 → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
1412, 13syl 17 . . . . . . . 8 (𝐴 ∈ ω → (∅ ∈ 𝐴 → suc ∅ ⊆ 𝐴))
15 df-1o 8113 . . . . . . . . 9 1o = suc ∅
1615sseq1i 3921 . . . . . . . 8 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
1714, 16syl6ibr 255 . . . . . . 7 (𝐴 ∈ ω → (∅ ∈ 𝐴 → 1o𝐴))
1817imp 411 . . . . . 6 ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → 1o𝐴)
1911, 18sylbi 220 . . . . 5 (𝐴N → 1o𝐴)
20 1onn 8276 . . . . . 6 1o ∈ ω
21 eleq1 2840 . . . . . . . . 9 (𝑥 = 1o → (𝑥N ↔ 1oN))
22 breq2 5037 . . . . . . . . 9 (𝑥 = 1o → (1o <N 𝑥 ↔ 1o <N 1o))
2321, 22anbi12d 634 . . . . . . . 8 (𝑥 = 1o → ((𝑥N ∧ 1o <N 𝑥) ↔ (1oN ∧ 1o <N 1o)))
2423, 5imbi12d 349 . . . . . . 7 (𝑥 = 1o → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((1oN ∧ 1o <N 1o) → 𝜓)))
25 eleq1 2840 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥N𝑦N))
26 breq2 5037 . . . . . . . . 9 (𝑥 = 𝑦 → (1o <N 𝑥 ↔ 1o <N 𝑦))
2725, 26anbi12d 634 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥N ∧ 1o <N 𝑥) ↔ (𝑦N ∧ 1o <N 𝑦)))
28 indpi.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
2927, 28imbi12d 349 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1o <N 𝑦) → 𝜒)))
30 pinn 10331 . . . . . . . . . . . . . . 15 (𝑥N𝑥 ∈ ω)
31 eleq1 2840 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ suc 𝑦 ∈ ω))
32 peano2b 7596 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
3331, 32bitr4di 293 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
3430, 33syl5ib 247 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝑥N𝑦 ∈ ω))
3534adantrd 496 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦 ∈ ω))
36 1pi 10336 . . . . . . . . . . . . . . . 16 1oN
37 ltpiord 10340 . . . . . . . . . . . . . . . 16 ((1oN𝑥N) → (1o <N 𝑥 ↔ 1o𝑥))
3836, 37mpan 690 . . . . . . . . . . . . . . 15 (𝑥N → (1o <N 𝑥 ↔ 1o𝑥))
3938biimpa 481 . . . . . . . . . . . . . 14 ((𝑥N ∧ 1o <N 𝑥) → 1o𝑥)
40 eleq2 2841 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → (1o𝑥 ↔ 1o ∈ suc 𝑦))
41 elsuci 6236 . . . . . . . . . . . . . . . 16 (1o ∈ suc 𝑦 → (1o𝑦 ∨ 1o = 𝑦))
42 ne0i 4234 . . . . . . . . . . . . . . . . 17 (1o𝑦𝑦 ≠ ∅)
43 0lt1o 8140 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ 1o
44 eleq2 2841 . . . . . . . . . . . . . . . . . . 19 (1o = 𝑦 → (∅ ∈ 1o ↔ ∅ ∈ 𝑦))
4543, 44mpbii 236 . . . . . . . . . . . . . . . . . 18 (1o = 𝑦 → ∅ ∈ 𝑦)
4645ne0d 4235 . . . . . . . . . . . . . . . . 17 (1o = 𝑦𝑦 ≠ ∅)
4742, 46jaoi 855 . . . . . . . . . . . . . . . 16 ((1o𝑦 ∨ 1o = 𝑦) → 𝑦 ≠ ∅)
4841, 47syl 17 . . . . . . . . . . . . . . 15 (1o ∈ suc 𝑦𝑦 ≠ ∅)
4940, 48syl6bi 256 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (1o𝑥𝑦 ≠ ∅))
5039, 49syl5 34 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦 ≠ ∅))
5135, 50jcad 517 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
52 elni 10329 . . . . . . . . . . . 12 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
5351, 52syl6ibr 255 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 𝑦N))
54 simpr 489 . . . . . . . . . . . 12 ((𝑥N ∧ 1o <N 𝑥) → 1o <N 𝑥)
55 breq2 5037 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (1o <N 𝑥 ↔ 1o <N suc 𝑦))
5654, 55syl5ib 247 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → 1o <N suc 𝑦))
5753, 56jcad 517 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) → (𝑦N ∧ 1o <N suc 𝑦)))
58 addclpi 10345 . . . . . . . . . . . . . . 15 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) ∈ N)
5936, 58mpan2 691 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1o) ∈ N)
60 addpiord 10337 . . . . . . . . . . . . . . . . . . 19 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) = (𝑦 +o 1o))
6136, 60mpan2 691 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +N 1o) = (𝑦 +o 1o))
62 pion 10332 . . . . . . . . . . . . . . . . . . 19 (𝑦N𝑦 ∈ On)
63 oa1suc 8167 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦N → (𝑦 +o 1o) = suc 𝑦)
6561, 64eqtrd 2794 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1o) = suc 𝑦)
6665eqeq2d 2770 . . . . . . . . . . . . . . . 16 (𝑦N → (𝑥 = (𝑦 +N 1o) ↔ 𝑥 = suc 𝑦))
6766biimparc 484 . . . . . . . . . . . . . . 15 ((𝑥 = suc 𝑦𝑦N) → 𝑥 = (𝑦 +N 1o))
6867eleq1d 2837 . . . . . . . . . . . . . 14 ((𝑥 = suc 𝑦𝑦N) → (𝑥N ↔ (𝑦 +N 1o) ∈ N))
6959, 68syl5ibr 249 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑦N) → (𝑦N𝑥N))
7069ex 417 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (𝑦N → (𝑦N𝑥N)))
7170pm2.43d 53 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝑦N𝑥N))
7255biimprd 251 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (1o <N suc 𝑦 → 1o <N 𝑥))
7371, 72anim12d 612 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝑦N ∧ 1o <N suc 𝑦) → (𝑥N ∧ 1o <N 𝑥)))
7457, 73impbid 215 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝑥N ∧ 1o <N 𝑥) ↔ (𝑦N ∧ 1o <N suc 𝑦)))
7574imbi1d 346 . . . . . . . 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 257 . . . . . . . . . . 11 (𝑦N → (𝑥 = suc 𝑦 → (𝜑𝜃)))
7877adantr 485 . . . . . . . . . 10 ((𝑦N ∧ 1o <N suc 𝑦) → (𝑥 = suc 𝑦 → (𝜑𝜃)))
7978com12 32 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝑦N ∧ 1o <N suc 𝑦) → (𝜑𝜃)))
8079pm5.74d 276 . . . . . . . 8 (𝑥 = suc 𝑦 → (((𝑦N ∧ 1o <N suc 𝑦) → 𝜑) ↔ ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
8175, 80bitrd 282 . . . . . . 7 (𝑥 = suc 𝑦 → (((𝑥N ∧ 1o <N 𝑥) → 𝜑) ↔ ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
82 eleq1 2840 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥N𝐴N))
83 breq2 5037 . . . . . . . . 9 (𝑥 = 𝐴 → (1o <N 𝑥 ↔ 1o <N 𝐴))
8482, 83anbi12d 634 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥N ∧ 1o <N 𝑥) ↔ (𝐴N ∧ 1o <N 𝐴)))
8584, 3imbi12d 349 . . . . . . 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 10340 . . . . . . . . . . . . . . 15 ((1oN𝑦N) → (1o <N 𝑦 ↔ 1o𝑦))
8836, 87mpan 690 . . . . . . . . . . . . . 14 (𝑦N → (1o <N 𝑦 ↔ 1o𝑦))
8988pm5.32i 579 . . . . . . . . . . . . 13 ((𝑦N ∧ 1o <N 𝑦) ↔ (𝑦N ∧ 1o𝑦))
9089simplbi2 505 . . . . . . . . . . . 12 (𝑦N → (1o𝑦 → (𝑦N ∧ 1o <N 𝑦)))
9190imim1d 82 . . . . . . . . . . 11 (𝑦N → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → (1o𝑦𝜒)))
92 ltrelpi 10342 . . . . . . . . . . . . . . 15 <N ⊆ (N × N)
9392brel 5587 . . . . . . . . . . . . . 14 (1o <N suc 𝑦 → (1oN ∧ suc 𝑦N))
94 ltpiord 10340 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . 12 (1o <N suc 𝑦 → 1o ∈ suc 𝑦)
971eqvinc 3561 . . . . . . . . . . . . . . 15 (1o = 𝑦 ↔ ∃𝑥(𝑥 = 1o𝑥 = 𝑦))
9897, 28, 6gencl 3451 . . . . . . . . . . . . . 14 (1o = 𝑦𝜒)
99 jao 959 . . . . . . . . . . . . . 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 415 . . . . . . . . 9 (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜒))
10515sseq1i 3921 . . . . . . . . . . 11 (1o𝑦 ↔ suc ∅ ⊆ 𝑦)
106 0ex 5178 . . . . . . . . . . . 12 ∅ ∈ V
107 sucssel 6262 . . . . . . . . . . . 12 (∅ ∈ V → (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦))
108106, 107ax-mp 5 . . . . . . . . . . 11 (suc ∅ ⊆ 𝑦 → ∅ ∈ 𝑦)
109105, 108sylbi 220 . . . . . . . . . 10 (1o𝑦 → ∅ ∈ 𝑦)
110 elni2 10330 . . . . . . . . . . 11 (𝑦N ↔ (𝑦 ∈ ω ∧ ∅ ∈ 𝑦))
111 indpi.6 . . . . . . . . . . 11 (𝑦N → (𝜒𝜃))
112110, 111sylbir 238 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∅ ∈ 𝑦) → (𝜒𝜃))
113109, 112sylan2 596 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 1o𝑦) → (𝜒𝜃))
114104, 113syl9r 78 . . . . . . . 8 ((𝑦 ∈ ω ∧ 1o𝑦) → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
115114adantlr 715 . . . . . . 7 (((𝑦 ∈ ω ∧ 1o ∈ ω) ∧ 1o𝑦) → (((𝑦N ∧ 1o <N 𝑦) → 𝜒) → ((𝑦N ∧ 1o <N suc 𝑦) → 𝜃)))
11624, 29, 81, 85, 86, 115findsg 7610 . . . . . 6 (((𝐴 ∈ ω ∧ 1o ∈ ω) ∧ 1o𝐴) → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
11720, 116mpanl2 701 . . . . 5 ((𝐴 ∈ ω ∧ 1o𝐴) → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
11810, 19, 117syl2anc 588 . . . 4 (𝐴N → ((𝐴N ∧ 1o <N 𝐴) → 𝜏))
119118expd 420 . . 3 (𝐴N → (𝐴N → (1o <N 𝐴𝜏)))
120119pm2.43i 52 . 2 (𝐴N → (1o <N 𝐴𝜏))
121 nlt1pi 10359 . . . 4 ¬ 𝐴 <N 1o
122 ltsopi 10341 . . . . . 6 <N Or N
123 sotric 5471 . . . . . 6 (( <N Or N ∧ (𝐴N ∧ 1oN)) → (𝐴 <N 1o ↔ ¬ (𝐴 = 1o ∨ 1o <N 𝐴)))
124122, 123mpan 690 . . . . 5 ((𝐴N ∧ 1oN) → (𝐴 <N 1o ↔ ¬ (𝐴 = 1o ∨ 1o <N 𝐴)))
12536, 124mpan2 691 . . . 4 (𝐴N → (𝐴 <N 1o ↔ ¬ (𝐴 = 1o ∨ 1o <N 𝐴)))
126121, 125mtbii 330 . . 3 (𝐴N → ¬ ¬ (𝐴 = 1o ∨ 1o <N 𝐴))
127126notnotrd 135 . 2 (𝐴N → (𝐴 = 1o ∨ 1o <N 𝐴))
1289, 120, 127mpjaod 858 1 (𝐴N𝜏)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  Vcvv 3410   ⊆ wss 3859  ∅c0 4226   class class class wbr 5033   Or wor 5443  Ord word 6169  Oncon0 6170  suc csuc 6172  (class class class)co 7151  ωcom 7580  1oc1o 8106   +o coa 8110  Ncnpi 10297   +N cpli 10298
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