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Theorem indpi 7143
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 7109 . . 3 (𝑥N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3 eqid 2137 . . . . . . . . . 10 ∅ = ∅
43orci 720 . . . . . . . . 9 (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5 nfv 1508 . . . . . . . . . . 11 𝑥∅ = ∅
6 nfsbc1v 2922 . . . . . . . . . . 11 𝑥[∅ / 𝑥]𝜑
75, 6nfor 1553 . . . . . . . . . 10 𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8 0ex 4050 . . . . . . . . . 10 ∅ ∈ V
9 eqeq1 2144 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10 sbceq1a 2913 . . . . . . . . . . 11 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
119, 10orbi12d 782 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
127, 8, 11elabf 2822 . . . . . . . . 9 (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
134, 12mpbir 145 . . . . . . . 8 ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14 suceq 4319 . . . . . . . . . . . . . 14 (𝑦 = ∅ → suc 𝑦 = suc ∅)
15 df-1o 6306 . . . . . . . . . . . . . 14 1o = suc ∅
1614, 15syl6eqr 2188 . . . . . . . . . . . . 13 (𝑦 = ∅ → suc 𝑦 = 1o)
17 indpi.5 . . . . . . . . . . . . . . 15 𝜓
1817olci 721 . . . . . . . . . . . . . 14 (1o = ∅ ∨ 𝜓)
19 1oex 6314 . . . . . . . . . . . . . . 15 1o ∈ V
20 eqeq1 2144 . . . . . . . . . . . . . . . 16 (𝑥 = 1o → (𝑥 = ∅ ↔ 1o = ∅))
21 indpi.1 . . . . . . . . . . . . . . . 16 (𝑥 = 1o → (𝜑𝜓))
2220, 21orbi12d 782 . . . . . . . . . . . . . . 15 (𝑥 = 1o → ((𝑥 = ∅ ∨ 𝜑) ↔ (1o = ∅ ∨ 𝜓)))
2319, 22elab 2823 . . . . . . . . . . . . . 14 (1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1o = ∅ ∨ 𝜓))
2418, 23mpbir 145 . . . . . . . . . . . . 13 1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
2516, 24syl6eqel 2228 . . . . . . . . . . . 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 7109 . . . . . . . . . . . . . . . 16 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
3029simprbi 273 . . . . . . . . . . . . . . 15 (𝑦N𝑦 ≠ ∅)
3130neneqd 2327 . . . . . . . . . . . . . 14 (𝑦N → ¬ 𝑦 = ∅)
32 biorf 733 . . . . . . . . . . . . . 14 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
3331, 32syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
34 vex 2684 . . . . . . . . . . . . . 14 𝑦 ∈ V
35 eqeq1 2144 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
36 indpi.2 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝜑𝜒))
3735, 36orbi12d 782 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
3834, 37elab 2823 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
3933, 38syl6bbr 197 . . . . . . . . . . . 12 (𝑦N → (𝜒𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
40 1pi 7116 . . . . . . . . . . . . . . . . . 18 1oN
41 addclpi 7128 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) ∈ N)
4240, 41mpan2 421 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1o) ∈ N)
43 elni 7109 . . . . . . . . . . . . . . . . 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 2327 . . . . . . . . . . . . . 14 (𝑦N → ¬ (𝑦 +N 1o) = ∅)
47 biorf 733 . . . . . . . . . . . . . 14 (¬ (𝑦 +N 1o) = ∅ → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
4846, 47syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
49 eqeq1 2144 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1o) → (𝑥 = ∅ ↔ (𝑦 +N 1o) = ∅))
50 indpi.3 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1o) → (𝜑𝜃))
5149, 50orbi12d 782 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 +N 1o) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
5251elabg 2825 . . . . . . . . . . . . . 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 7117 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ 1oN) → (𝑦 +N 1o) = (𝑦 +o 1o))
5540, 54mpan2 421 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1o) = (𝑦 +o 1o))
56 pion 7111 . . . . . . . . . . . . . . . 16 (𝑦N𝑦 ∈ On)
57 oa1suc 6356 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
5856, 57syl 14 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +o 1o) = suc 𝑦)
5955, 58eqtrd 2170 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1o) = suc 𝑦)
6059eleq1d 2206 . . . . . . . . . . . . 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 4526 . . . . . . . . . . . 12 (𝑦 ∈ ω → DECID 𝑦 = ∅)
65 df-dc 820 . . . . . . . . . . . 12 (DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
6664, 65sylib 121 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
67 idd 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅))
6867necon3bd 2349 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ≠ ∅))
6968anc2li 327 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
7069, 29syl6ibr 161 . . . . . . . . . . . 12 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦N))
7170orim2d 777 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦N)))
7266, 71mpd 13 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦N))
7327, 63, 72mpjaod 707 . . . . . . . . 9 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
7473rgen 2483 . . . . . . . 8 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
75 peano5 4507 . . . . . . . 8 ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
7613, 74, 75mp2an 422 . . . . . . 7 ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
7776sseli 3088 . . . . . 6 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
78 abid 2125 . . . . . 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 2307 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
82 biorf 733 . . . . . 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 2747 1 (𝐴N𝜏)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  DECID wdc 819   = wceq 1331  wcel 1480  {cab 2123  wne 2306  wral 2414  [wsbc 2904  wss 3066  c0 3358  Oncon0 4280  suc csuc 4282  ωcom 4499  (class class class)co 5767  1oc1o 6299   +o coa 6303  Ncnpi 7073   +N cpli 7074
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-ni 7105  df-pli 7106
This theorem is referenced by:  pitonn  7649
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