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Theorem indpi 7257
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 7223 . . 3 (𝑥N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3 eqid 2157 . . . . . . . . . 10 ∅ = ∅
43orci 721 . . . . . . . . 9 (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5 nfv 1508 . . . . . . . . . . 11 𝑥∅ = ∅
6 nfsbc1v 2955 . . . . . . . . . . 11 𝑥[∅ / 𝑥]𝜑
75, 6nfor 1554 . . . . . . . . . 10 𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8 0ex 4091 . . . . . . . . . 10 ∅ ∈ V
9 eqeq1 2164 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10 sbceq1a 2946 . . . . . . . . . . 11 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
119, 10orbi12d 783 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
127, 8, 11elabf 2855 . . . . . . . . 9 (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
134, 12mpbir 145 . . . . . . . 8 ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14 suceq 4362 . . . . . . . . . . . . . 14 (𝑦 = ∅ → suc 𝑦 = suc ∅)
15 df-1o 6360 . . . . . . . . . . . . . 14 1o = suc ∅
1614, 15eqtr4di 2208 . . . . . . . . . . . . 13 (𝑦 = ∅ → suc 𝑦 = 1o)
17 indpi.5 . . . . . . . . . . . . . . 15 𝜓
1817olci 722 . . . . . . . . . . . . . 14 (1o = ∅ ∨ 𝜓)
19 1oex 6368 . . . . . . . . . . . . . . 15 1o ∈ V
20 eqeq1 2164 . . . . . . . . . . . . . . . 16 (𝑥 = 1o → (𝑥 = ∅ ↔ 1o = ∅))
21 indpi.1 . . . . . . . . . . . . . . . 16 (𝑥 = 1o → (𝜑𝜓))
2220, 21orbi12d 783 . . . . . . . . . . . . . . 15 (𝑥 = 1o → ((𝑥 = ∅ ∨ 𝜑) ↔ (1o = ∅ ∨ 𝜓)))
2319, 22elab 2856 . . . . . . . . . . . . . 14 (1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1o = ∅ ∨ 𝜓))
2418, 23mpbir 145 . . . . . . . . . . . . 13 1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
2516, 24eqeltrdi 2248 . . . . . . . . . . . 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 7223 . . . . . . . . . . . . . . . 16 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
3029simprbi 273 . . . . . . . . . . . . . . 15 (𝑦N𝑦 ≠ ∅)
3130neneqd 2348 . . . . . . . . . . . . . 14 (𝑦N → ¬ 𝑦 = ∅)
32 biorf 734 . . . . . . . . . . . . . 14 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
3331, 32syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
34 vex 2715 . . . . . . . . . . . . . 14 𝑦 ∈ V
35 eqeq1 2164 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
36 indpi.2 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝜑𝜒))
3735, 36orbi12d 783 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
3834, 37elab 2856 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
3933, 38bitr4di 197 . . . . . . . . . . . 12 (𝑦N → (𝜒𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
40 1pi 7230 . . . . . . . . . . . . . . . . . 18 1oN
41 addclpi 7242 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) ∈ N)
4240, 41mpan2 422 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1o) ∈ N)
43 elni 7223 . . . . . . . . . . . . . . . . 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 2348 . . . . . . . . . . . . . 14 (𝑦N → ¬ (𝑦 +N 1o) = ∅)
47 biorf 734 . . . . . . . . . . . . . 14 (¬ (𝑦 +N 1o) = ∅ → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
4846, 47syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
49 eqeq1 2164 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1o) → (𝑥 = ∅ ↔ (𝑦 +N 1o) = ∅))
50 indpi.3 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1o) → (𝜑𝜃))
5149, 50orbi12d 783 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 +N 1o) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
5251elabg 2858 . . . . . . . . . . . . . 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 7231 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) = (𝑦 +o 1o))
5540, 54mpan2 422 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1o) = (𝑦 +o 1o))
56 pion 7225 . . . . . . . . . . . . . . . 16 (𝑦N𝑦 ∈ On)
57 oa1suc 6411 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
5856, 57syl 14 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +o 1o) = suc 𝑦)
5955, 58eqtrd 2190 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1o) = suc 𝑦)
6059eleq1d 2226 . . . . . . . . . . . . 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 4576 . . . . . . . . . . . 12 (𝑦 ∈ ω → DECID 𝑦 = ∅)
65 df-dc 821 . . . . . . . . . . . 12 (DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
6664, 65sylib 121 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
67 idd 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅))
6867necon3bd 2370 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ≠ ∅))
6968anc2li 327 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
7069, 29syl6ibr 161 . . . . . . . . . . . 12 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦N))
7170orim2d 778 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦N)))
7266, 71mpd 13 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦N))
7327, 63, 72mpjaod 708 . . . . . . . . 9 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
7473rgen 2510 . . . . . . . 8 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
75 peano5 4556 . . . . . . . 8 ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
7613, 74, 75mp2an 423 . . . . . . 7 ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
7776sseli 3124 . . . . . 6 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
78 abid 2145 . . . . . 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 2328 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
82 biorf 734 . . . . . 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 2778 1 (𝐴N𝜏)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 820   = wceq 1335  wcel 2128  {cab 2143  wne 2327  wral 2435  [wsbc 2937  wss 3102  c0 3394  Oncon0 4323  suc csuc 4325  ωcom 4548  (class class class)co 5821  1oc1o 6353   +o coa 6357  Ncnpi 7187   +N cpli 7188
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-1o 6360  df-oadd 6364  df-ni 7219  df-pli 7220
This theorem is referenced by:  pitonn  7763
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