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Theorem indpi 6440
 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 6406 . . 3 (𝑥N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3 eqid 2040 . . . . . . . . . 10 ∅ = ∅
43orci 650 . . . . . . . . 9 (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5 nfv 1421 . . . . . . . . . . 11 𝑥∅ = ∅
6 nfsbc1v 2782 . . . . . . . . . . 11 𝑥[∅ / 𝑥]𝜑
75, 6nfor 1466 . . . . . . . . . 10 𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8 0ex 3884 . . . . . . . . . 10 ∅ ∈ V
9 eqeq1 2046 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10 sbceq1a 2773 . . . . . . . . . . 11 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
119, 10orbi12d 707 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
127, 8, 11elabf 2686 . . . . . . . . 9 (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
134, 12mpbir 134 . . . . . . . 8 ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14 suceq 4139 . . . . . . . . . . . . . 14 (𝑦 = ∅ → suc 𝑦 = suc ∅)
15 df-1o 6001 . . . . . . . . . . . . . 14 1𝑜 = suc ∅
1614, 15syl6eqr 2090 . . . . . . . . . . . . 13 (𝑦 = ∅ → suc 𝑦 = 1𝑜)
17 indpi.5 . . . . . . . . . . . . . . 15 𝜓
1817olci 651 . . . . . . . . . . . . . 14 (1𝑜 = ∅ ∨ 𝜓)
19 1onn 6093 . . . . . . . . . . . . . . . 16 1𝑜 ∈ ω
2019elexi 2567 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
21 eqeq1 2046 . . . . . . . . . . . . . . . 16 (𝑥 = 1𝑜 → (𝑥 = ∅ ↔ 1𝑜 = ∅))
22 indpi.1 . . . . . . . . . . . . . . . 16 (𝑥 = 1𝑜 → (𝜑𝜓))
2321, 22orbi12d 707 . . . . . . . . . . . . . . 15 (𝑥 = 1𝑜 → ((𝑥 = ∅ ∨ 𝜑) ↔ (1𝑜 = ∅ ∨ 𝜓)))
2420, 23elab 2687 . . . . . . . . . . . . . 14 (1𝑜 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1𝑜 = ∅ ∨ 𝜓))
2518, 24mpbir 134 . . . . . . . . . . . . 13 1𝑜 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
2616, 25syl6eqel 2128 . . . . . . . . . . . 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 6406 . . . . . . . . . . . . . . . 16 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
3130simprbi 260 . . . . . . . . . . . . . . 15 (𝑦N𝑦 ≠ ∅)
3231neneqd 2226 . . . . . . . . . . . . . 14 (𝑦N → ¬ 𝑦 = ∅)
33 biorf 663 . . . . . . . . . . . . . 14 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
3432, 33syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
35 vex 2560 . . . . . . . . . . . . . 14 𝑦 ∈ V
36 eqeq1 2046 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
37 indpi.2 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝜑𝜒))
3836, 37orbi12d 707 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
3935, 38elab 2687 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
4034, 39syl6bbr 187 . . . . . . . . . . . 12 (𝑦N → (𝜒𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
41 1pi 6413 . . . . . . . . . . . . . . . . . 18 1𝑜N
42 addclpi 6425 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) ∈ N)
4341, 42mpan2 401 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1𝑜) ∈ N)
44 elni 6406 . . . . . . . . . . . . . . . . 17 ((𝑦 +N 1𝑜) ∈ N ↔ ((𝑦 +N 1𝑜) ∈ ω ∧ (𝑦 +N 1𝑜) ≠ ∅))
4543, 44sylib 127 . . . . . . . . . . . . . . . 16 (𝑦N → ((𝑦 +N 1𝑜) ∈ ω ∧ (𝑦 +N 1𝑜) ≠ ∅))
4645simprd 107 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1𝑜) ≠ ∅)
4746neneqd 2226 . . . . . . . . . . . . . 14 (𝑦N → ¬ (𝑦 +N 1𝑜) = ∅)
48 biorf 663 . . . . . . . . . . . . . 14 (¬ (𝑦 +N 1𝑜) = ∅ → (𝜃 ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
4947, 48syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜃 ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
50 eqeq1 2046 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1𝑜) → (𝑥 = ∅ ↔ (𝑦 +N 1𝑜) = ∅))
51 indpi.3 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1𝑜) → (𝜑𝜃))
5250, 51orbi12d 707 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 +N 1𝑜) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
5352elabg 2688 . . . . . . . . . . . . . 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 6414 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
5641, 55mpan2 401 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
57 pion 6408 . . . . . . . . . . . . . . . 16 (𝑦N𝑦 ∈ On)
58 oa1suc 6047 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
5957, 58syl 14 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +𝑜 1𝑜) = suc 𝑦)
6056, 59eqtrd 2072 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1𝑜) = suc 𝑦)
6160eleq1d 2106 . . . . . . . . . . . . 13 (𝑦N → ((𝑦 +N 1𝑜) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6249, 54, 613bitr2d 205 . . . . . . . . . . . 12 (𝑦N → (𝜃 ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6329, 40, 623imtr3d 191 . . . . . . . . . . 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 4339 . . . . . . . . . . . 12 (𝑦 ∈ ω → DECID 𝑦 = ∅)
66 df-dc 743 . . . . . . . . . . . 12 (DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
6765, 66sylib 127 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
68 idd 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅))
6968necon3bd 2248 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ≠ ∅))
7069anc2li 312 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
7170, 30syl6ibr 151 . . . . . . . . . . . 12 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦N))
7271orim2d 702 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦N)))
7367, 72mpd 13 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦N))
7428, 64, 73mpjaod 638 . . . . . . . . 9 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
7574rgen 2374 . . . . . . . 8 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
76 peano5 4321 . . . . . . . 8 ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
7713, 75, 76mp2an 402 . . . . . . 7 ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
7877sseli 2941 . . . . . 6 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
79 abid 2028 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑥 = ∅ ∨ 𝜑))
8078, 79sylib 127 . . . . 5 (𝑥 ∈ ω → (𝑥 = ∅ ∨ 𝜑))
8180adantr 261 . . . 4 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝑥 = ∅ ∨ 𝜑))
82 df-ne 2206 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
83 biorf 663 . . . . . 6 𝑥 = ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8482, 83sylbi 114 . . . . 5 (𝑥 ≠ ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8584adantl 262 . . . 4 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8681, 85mpbird 156 . . 3 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → 𝜑)
872, 86sylbi 114 . 2 (𝑥N𝜑)
881, 87vtoclga 2619 1 (𝐴N𝜏)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742   = wceq 1243   ∈ wcel 1393  {cab 2026   ≠ wne 2204  ∀wral 2306  [wsbc 2764   ⊆ wss 2917  ∅c0 3224  Oncon0 4100  suc csuc 4102  ωcom 4313  (class class class)co 5512  1𝑜c1o 5994   +𝑜 coa 5998  Ncnpi 6370   +N cpli 6371 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-ni 6402  df-pli 6403 This theorem is referenced by:  pitonn  6924
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