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Theorem indpi 6498
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 6464 . . 3 (𝑥N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3 eqid 2056 . . . . . . . . . 10 ∅ = ∅
43orci 660 . . . . . . . . 9 (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5 nfv 1437 . . . . . . . . . . 11 𝑥∅ = ∅
6 nfsbc1v 2805 . . . . . . . . . . 11 𝑥[∅ / 𝑥]𝜑
75, 6nfor 1482 . . . . . . . . . 10 𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8 0ex 3912 . . . . . . . . . 10 ∅ ∈ V
9 eqeq1 2062 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10 sbceq1a 2796 . . . . . . . . . . 11 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
119, 10orbi12d 717 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
127, 8, 11elabf 2709 . . . . . . . . 9 (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
134, 12mpbir 138 . . . . . . . 8 ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14 suceq 4167 . . . . . . . . . . . . . 14 (𝑦 = ∅ → suc 𝑦 = suc ∅)
15 df-1o 6032 . . . . . . . . . . . . . 14 1𝑜 = suc ∅
1614, 15syl6eqr 2106 . . . . . . . . . . . . 13 (𝑦 = ∅ → suc 𝑦 = 1𝑜)
17 indpi.5 . . . . . . . . . . . . . . 15 𝜓
1817olci 661 . . . . . . . . . . . . . 14 (1𝑜 = ∅ ∨ 𝜓)
19 1onn 6124 . . . . . . . . . . . . . . . 16 1𝑜 ∈ ω
2019elexi 2584 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
21 eqeq1 2062 . . . . . . . . . . . . . . . 16 (𝑥 = 1𝑜 → (𝑥 = ∅ ↔ 1𝑜 = ∅))
22 indpi.1 . . . . . . . . . . . . . . . 16 (𝑥 = 1𝑜 → (𝜑𝜓))
2321, 22orbi12d 717 . . . . . . . . . . . . . . 15 (𝑥 = 1𝑜 → ((𝑥 = ∅ ∨ 𝜑) ↔ (1𝑜 = ∅ ∨ 𝜓)))
2420, 23elab 2710 . . . . . . . . . . . . . 14 (1𝑜 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1𝑜 = ∅ ∨ 𝜓))
2518, 24mpbir 138 . . . . . . . . . . . . 13 1𝑜 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
2616, 25syl6eqel 2144 . . . . . . . . . . . 12 (𝑦 = ∅ → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
2726a1d 22 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
2827a1i 9 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 = ∅ → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})))
29 indpi.6 . . . . . . . . . . . 12 (𝑦N → (𝜒𝜃))
30 elni 6464 . . . . . . . . . . . . . . . 16 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
3130simprbi 264 . . . . . . . . . . . . . . 15 (𝑦N𝑦 ≠ ∅)
3231neneqd 2241 . . . . . . . . . . . . . 14 (𝑦N → ¬ 𝑦 = ∅)
33 biorf 673 . . . . . . . . . . . . . 14 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
3432, 33syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
35 vex 2577 . . . . . . . . . . . . . 14 𝑦 ∈ V
36 eqeq1 2062 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
37 indpi.2 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝜑𝜒))
3836, 37orbi12d 717 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
3935, 38elab 2710 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
4034, 39syl6bbr 191 . . . . . . . . . . . 12 (𝑦N → (𝜒𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
41 1pi 6471 . . . . . . . . . . . . . . . . . 18 1𝑜N
42 addclpi 6483 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) ∈ N)
4341, 42mpan2 409 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1𝑜) ∈ N)
44 elni 6464 . . . . . . . . . . . . . . . . 17 ((𝑦 +N 1𝑜) ∈ N ↔ ((𝑦 +N 1𝑜) ∈ ω ∧ (𝑦 +N 1𝑜) ≠ ∅))
4543, 44sylib 131 . . . . . . . . . . . . . . . 16 (𝑦N → ((𝑦 +N 1𝑜) ∈ ω ∧ (𝑦 +N 1𝑜) ≠ ∅))
4645simprd 111 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1𝑜) ≠ ∅)
4746neneqd 2241 . . . . . . . . . . . . . 14 (𝑦N → ¬ (𝑦 +N 1𝑜) = ∅)
48 biorf 673 . . . . . . . . . . . . . 14 (¬ (𝑦 +N 1𝑜) = ∅ → (𝜃 ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
4947, 48syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜃 ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
50 eqeq1 2062 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1𝑜) → (𝑥 = ∅ ↔ (𝑦 +N 1𝑜) = ∅))
51 indpi.3 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1𝑜) → (𝜑𝜃))
5250, 51orbi12d 717 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 +N 1𝑜) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
5352elabg 2711 . . . . . . . . . . . . . 14 ((𝑦 +N 1𝑜) ∈ N → ((𝑦 +N 1𝑜) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
5443, 53syl 14 . . . . . . . . . . . . 13 (𝑦N → ((𝑦 +N 1𝑜) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
55 addpiord 6472 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
5641, 55mpan2 409 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
57 pion 6466 . . . . . . . . . . . . . . . 16 (𝑦N𝑦 ∈ On)
58 oa1suc 6078 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
5957, 58syl 14 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +𝑜 1𝑜) = suc 𝑦)
6056, 59eqtrd 2088 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1𝑜) = suc 𝑦)
6160eleq1d 2122 . . . . . . . . . . . . 13 (𝑦N → ((𝑦 +N 1𝑜) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6249, 54, 613bitr2d 209 . . . . . . . . . . . 12 (𝑦N → (𝜃 ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6329, 40, 623imtr3d 195 . . . . . . . . . . 11 (𝑦N → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6463a1i 9 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦N → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})))
65 nndceq0 4367 . . . . . . . . . . . 12 (𝑦 ∈ ω → DECID 𝑦 = ∅)
66 df-dc 754 . . . . . . . . . . . 12 (DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
6765, 66sylib 131 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
68 idd 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅))
6968necon3bd 2263 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ≠ ∅))
7069anc2li 316 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
7170, 30syl6ibr 155 . . . . . . . . . . . 12 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦N))
7271orim2d 712 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦N)))
7367, 72mpd 13 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦N))
7428, 64, 73mpjaod 648 . . . . . . . . 9 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
7574rgen 2391 . . . . . . . 8 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
76 peano5 4349 . . . . . . . 8 ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
7713, 75, 76mp2an 410 . . . . . . 7 ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
7877sseli 2969 . . . . . 6 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
79 abid 2044 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑥 = ∅ ∨ 𝜑))
8078, 79sylib 131 . . . . 5 (𝑥 ∈ ω → (𝑥 = ∅ ∨ 𝜑))
8180adantr 265 . . . 4 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝑥 = ∅ ∨ 𝜑))
82 df-ne 2221 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
83 biorf 673 . . . . . 6 𝑥 = ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8482, 83sylbi 118 . . . . 5 (𝑥 ≠ ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8584adantl 266 . . . 4 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8681, 85mpbird 160 . . 3 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → 𝜑)
872, 86sylbi 118 . 2 (𝑥N𝜑)
881, 87vtoclga 2636 1 (𝐴N𝜏)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  DECID wdc 753   = wceq 1259  wcel 1409  {cab 2042  wne 2220  wral 2323  [wsbc 2787  wss 2945  c0 3252  Oncon0 4128  suc csuc 4130  ωcom 4341  (class class class)co 5540  1𝑜c1o 6025   +𝑜 coa 6029  Ncnpi 6428   +N cpli 6429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-ni 6460  df-pli 6461
This theorem is referenced by:  pitonn  6982
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