ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  indpi GIF version

Theorem indpi 7304
Description: Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)
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 indpi.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
2 elni 7270 . . 3 (𝑥N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3 eqid 2170 . . . . . . . . . 10 ∅ = ∅
43orci 726 . . . . . . . . 9 (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5 nfv 1521 . . . . . . . . . . 11 𝑥∅ = ∅
6 nfsbc1v 2973 . . . . . . . . . . 11 𝑥[∅ / 𝑥]𝜑
75, 6nfor 1567 . . . . . . . . . 10 𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8 0ex 4116 . . . . . . . . . 10 ∅ ∈ V
9 eqeq1 2177 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10 sbceq1a 2964 . . . . . . . . . . 11 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
119, 10orbi12d 788 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
127, 8, 11elabf 2873 . . . . . . . . 9 (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
134, 12mpbir 145 . . . . . . . 8 ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14 suceq 4387 . . . . . . . . . . . . . 14 (𝑦 = ∅ → suc 𝑦 = suc ∅)
15 df-1o 6395 . . . . . . . . . . . . . 14 1o = suc ∅
1614, 15eqtr4di 2221 . . . . . . . . . . . . 13 (𝑦 = ∅ → suc 𝑦 = 1o)
17 indpi.5 . . . . . . . . . . . . . . 15 𝜓
1817olci 727 . . . . . . . . . . . . . 14 (1o = ∅ ∨ 𝜓)
19 1oex 6403 . . . . . . . . . . . . . . 15 1o ∈ V
20 eqeq1 2177 . . . . . . . . . . . . . . . 16 (𝑥 = 1o → (𝑥 = ∅ ↔ 1o = ∅))
21 indpi.1 . . . . . . . . . . . . . . . 16 (𝑥 = 1o → (𝜑𝜓))
2220, 21orbi12d 788 . . . . . . . . . . . . . . 15 (𝑥 = 1o → ((𝑥 = ∅ ∨ 𝜑) ↔ (1o = ∅ ∨ 𝜓)))
2319, 22elab 2874 . . . . . . . . . . . . . 14 (1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1o = ∅ ∨ 𝜓))
2418, 23mpbir 145 . . . . . . . . . . . . 13 1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
2516, 24eqeltrdi 2261 . . . . . . . . . . . 12 (𝑦 = ∅ → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
2625a1d 22 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
2726a1i 9 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 = ∅ → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})))
28 indpi.6 . . . . . . . . . . . 12 (𝑦N → (𝜒𝜃))
29 elni 7270 . . . . . . . . . . . . . . . 16 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
3029simprbi 273 . . . . . . . . . . . . . . 15 (𝑦N𝑦 ≠ ∅)
3130neneqd 2361 . . . . . . . . . . . . . 14 (𝑦N → ¬ 𝑦 = ∅)
32 biorf 739 . . . . . . . . . . . . . 14 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
3331, 32syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
34 vex 2733 . . . . . . . . . . . . . 14 𝑦 ∈ V
35 eqeq1 2177 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
36 indpi.2 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝜑𝜒))
3735, 36orbi12d 788 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
3834, 37elab 2874 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
3933, 38bitr4di 197 . . . . . . . . . . . 12 (𝑦N → (𝜒𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
40 1pi 7277 . . . . . . . . . . . . . . . . . 18 1oN
41 addclpi 7289 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) ∈ N)
4240, 41mpan2 423 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1o) ∈ N)
43 elni 7270 . . . . . . . . . . . . . . . . 17 ((𝑦 +N 1o) ∈ N ↔ ((𝑦 +N 1o) ∈ ω ∧ (𝑦 +N 1o) ≠ ∅))
4442, 43sylib 121 . . . . . . . . . . . . . . . 16 (𝑦N → ((𝑦 +N 1o) ∈ ω ∧ (𝑦 +N 1o) ≠ ∅))
4544simprd 113 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1o) ≠ ∅)
4645neneqd 2361 . . . . . . . . . . . . . 14 (𝑦N → ¬ (𝑦 +N 1o) = ∅)
47 biorf 739 . . . . . . . . . . . . . 14 (¬ (𝑦 +N 1o) = ∅ → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
4846, 47syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
49 eqeq1 2177 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1o) → (𝑥 = ∅ ↔ (𝑦 +N 1o) = ∅))
50 indpi.3 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1o) → (𝜑𝜃))
5149, 50orbi12d 788 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 +N 1o) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
5251elabg 2876 . . . . . . . . . . . . . 14 ((𝑦 +N 1o) ∈ N → ((𝑦 +N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
5342, 52syl 14 . . . . . . . . . . . . 13 (𝑦N → ((𝑦 +N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
54 addpiord 7278 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) = (𝑦 +o 1o))
5540, 54mpan2 423 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1o) = (𝑦 +o 1o))
56 pion 7272 . . . . . . . . . . . . . . . 16 (𝑦N𝑦 ∈ On)
57 oa1suc 6446 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
5856, 57syl 14 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +o 1o) = suc 𝑦)
5955, 58eqtrd 2203 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1o) = suc 𝑦)
6059eleq1d 2239 . . . . . . . . . . . . 13 (𝑦N → ((𝑦 +N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6148, 53, 603bitr2d 215 . . . . . . . . . . . 12 (𝑦N → (𝜃 ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6228, 39, 613imtr3d 201 . . . . . . . . . . 11 (𝑦N → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6362a1i 9 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦N → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})))
64 nndceq0 4602 . . . . . . . . . . . 12 (𝑦 ∈ ω → DECID 𝑦 = ∅)
65 df-dc 830 . . . . . . . . . . . 12 (DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
6664, 65sylib 121 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
67 idd 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅))
6867necon3bd 2383 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ≠ ∅))
6968anc2li 327 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
7069, 29syl6ibr 161 . . . . . . . . . . . 12 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦N))
7170orim2d 783 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦N)))
7266, 71mpd 13 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦N))
7327, 63, 72mpjaod 713 . . . . . . . . 9 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
7473rgen 2523 . . . . . . . 8 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
75 peano5 4582 . . . . . . . 8 ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
7613, 74, 75mp2an 424 . . . . . . 7 ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
7776sseli 3143 . . . . . 6 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
78 abid 2158 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑥 = ∅ ∨ 𝜑))
7977, 78sylib 121 . . . . 5 (𝑥 ∈ ω → (𝑥 = ∅ ∨ 𝜑))
8079adantr 274 . . . 4 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝑥 = ∅ ∨ 𝜑))
81 df-ne 2341 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
82 biorf 739 . . . . . 6 𝑥 = ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8381, 82sylbi 120 . . . . 5 (𝑥 ≠ ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8483adantl 275 . . . 4 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8580, 84mpbird 166 . . 3 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → 𝜑)
862, 85sylbi 120 . 2 (𝑥N𝜑)
871, 86vtoclga 2796 1 (𝐴N𝜏)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 703  DECID wdc 829   = wceq 1348  wcel 2141  {cab 2156  wne 2340  wral 2448  [wsbc 2955  wss 3121  c0 3414  Oncon0 4348  suc csuc 4350  ωcom 4574  (class class class)co 5853  1oc1o 6388   +o coa 6392  Ncnpi 7234   +N cpli 7235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-ni 7266  df-pli 7267
This theorem is referenced by:  pitonn  7810
  Copyright terms: Public domain W3C validator