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Theorem indpi 6880
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 6846 . . 3 (𝑥N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3 eqid 2088 . . . . . . . . . 10 ∅ = ∅
43orci 685 . . . . . . . . 9 (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5 nfv 1466 . . . . . . . . . . 11 𝑥∅ = ∅
6 nfsbc1v 2856 . . . . . . . . . . 11 𝑥[∅ / 𝑥]𝜑
75, 6nfor 1511 . . . . . . . . . 10 𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8 0ex 3958 . . . . . . . . . 10 ∅ ∈ V
9 eqeq1 2094 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10 sbceq1a 2847 . . . . . . . . . . 11 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
119, 10orbi12d 742 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
127, 8, 11elabf 2757 . . . . . . . . 9 (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
134, 12mpbir 144 . . . . . . . 8 ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14 suceq 4220 . . . . . . . . . . . . . 14 (𝑦 = ∅ → suc 𝑦 = suc ∅)
15 df-1o 6163 . . . . . . . . . . . . . 14 1𝑜 = suc ∅
1614, 15syl6eqr 2138 . . . . . . . . . . . . 13 (𝑦 = ∅ → suc 𝑦 = 1𝑜)
17 indpi.5 . . . . . . . . . . . . . . 15 𝜓
1817olci 686 . . . . . . . . . . . . . 14 (1𝑜 = ∅ ∨ 𝜓)
19 1oex 6171 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
20 eqeq1 2094 . . . . . . . . . . . . . . . 16 (𝑥 = 1𝑜 → (𝑥 = ∅ ↔ 1𝑜 = ∅))
21 indpi.1 . . . . . . . . . . . . . . . 16 (𝑥 = 1𝑜 → (𝜑𝜓))
2220, 21orbi12d 742 . . . . . . . . . . . . . . 15 (𝑥 = 1𝑜 → ((𝑥 = ∅ ∨ 𝜑) ↔ (1𝑜 = ∅ ∨ 𝜓)))
2319, 22elab 2758 . . . . . . . . . . . . . 14 (1𝑜 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1𝑜 = ∅ ∨ 𝜓))
2418, 23mpbir 144 . . . . . . . . . . . . 13 1𝑜 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
2516, 24syl6eqel 2178 . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . 16 (𝑦N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
3029simprbi 269 . . . . . . . . . . . . . . 15 (𝑦N𝑦 ≠ ∅)
3130neneqd 2276 . . . . . . . . . . . . . 14 (𝑦N → ¬ 𝑦 = ∅)
32 biorf 698 . . . . . . . . . . . . . 14 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
3331, 32syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
34 vex 2622 . . . . . . . . . . . . . 14 𝑦 ∈ V
35 eqeq1 2094 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
36 indpi.2 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝜑𝜒))
3735, 36orbi12d 742 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
3834, 37elab 2758 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
3933, 38syl6bbr 196 . . . . . . . . . . . 12 (𝑦N → (𝜒𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
40 1pi 6853 . . . . . . . . . . . . . . . . . 18 1𝑜N
41 addclpi 6865 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) ∈ N)
4240, 41mpan2 416 . . . . . . . . . . . . . . . . 17 (𝑦N → (𝑦 +N 1𝑜) ∈ N)
43 elni 6846 . . . . . . . . . . . . . . . . 17 ((𝑦 +N 1𝑜) ∈ N ↔ ((𝑦 +N 1𝑜) ∈ ω ∧ (𝑦 +N 1𝑜) ≠ ∅))
4442, 43sylib 120 . . . . . . . . . . . . . . . 16 (𝑦N → ((𝑦 +N 1𝑜) ∈ ω ∧ (𝑦 +N 1𝑜) ≠ ∅))
4544simprd 112 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1𝑜) ≠ ∅)
4645neneqd 2276 . . . . . . . . . . . . . 14 (𝑦N → ¬ (𝑦 +N 1𝑜) = ∅)
47 biorf 698 . . . . . . . . . . . . . 14 (¬ (𝑦 +N 1𝑜) = ∅ → (𝜃 ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
4846, 47syl 14 . . . . . . . . . . . . 13 (𝑦N → (𝜃 ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
49 eqeq1 2094 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1𝑜) → (𝑥 = ∅ ↔ (𝑦 +N 1𝑜) = ∅))
50 indpi.3 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 +N 1𝑜) → (𝜑𝜃))
5149, 50orbi12d 742 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 +N 1𝑜) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1𝑜) = ∅ ∨ 𝜃)))
5251elabg 2759 . . . . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ 1𝑜N) → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
5540, 54mpan2 416 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +N 1𝑜) = (𝑦 +𝑜 1𝑜))
56 pion 6848 . . . . . . . . . . . . . . . 16 (𝑦N𝑦 ∈ On)
57 oa1suc 6210 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
5856, 57syl 14 . . . . . . . . . . . . . . 15 (𝑦N → (𝑦 +𝑜 1𝑜) = suc 𝑦)
5955, 58eqtrd 2120 . . . . . . . . . . . . . 14 (𝑦N → (𝑦 +N 1𝑜) = suc 𝑦)
6059eleq1d 2156 . . . . . . . . . . . . 13 (𝑦N → ((𝑦 +N 1𝑜) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6148, 53, 603bitr2d 214 . . . . . . . . . . . 12 (𝑦N → (𝜃 ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
6228, 39, 613imtr3d 200 . . . . . . . . . . 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 4421 . . . . . . . . . . . 12 (𝑦 ∈ ω → DECID 𝑦 = ∅)
65 df-dc 781 . . . . . . . . . . . 12 (DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
6664, 65sylib 120 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
67 idd 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅))
6867necon3bd 2298 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ≠ ∅))
6968anc2li 322 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
7069, 29syl6ibr 160 . . . . . . . . . . . 12 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦N))
7170orim2d 737 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦N)))
7266, 71mpd 13 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦N))
7327, 63, 72mpjaod 673 . . . . . . . . 9 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
7473rgen 2428 . . . . . . . 8 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
75 peano5 4403 . . . . . . . 8 ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
7613, 74, 75mp2an 417 . . . . . . 7 ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
7776sseli 3019 . . . . . 6 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
78 abid 2076 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑥 = ∅ ∨ 𝜑))
7977, 78sylib 120 . . . . 5 (𝑥 ∈ ω → (𝑥 = ∅ ∨ 𝜑))
8079adantr 270 . . . 4 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝑥 = ∅ ∨ 𝜑))
81 df-ne 2256 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
82 biorf 698 . . . . . 6 𝑥 = ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8381, 82sylbi 119 . . . . 5 (𝑥 ≠ ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8483adantl 271 . . . 4 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
8580, 84mpbird 165 . . 3 ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → 𝜑)
862, 85sylbi 119 . 2 (𝑥N𝜑)
871, 86vtoclga 2685 1 (𝐴N𝜏)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  DECID wdc 780   = wceq 1289  wcel 1438  {cab 2074  wne 2255  wral 2359  [wsbc 2838  wss 2997  c0 3284  Oncon0 4181  suc csuc 4183  ωcom 4395  (class class class)co 5634  1𝑜c1o 6156   +𝑜 coa 6160  Ncnpi 6810   +N cpli 6811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-ni 6842  df-pli 6843
This theorem is referenced by:  pitonn  7364
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