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Theorem peano5 4375
Description: The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as theorem findes 4380. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
peano5 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfom3 4369 . . 3 ω = {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}
2 peano1 4371 . . . . . . . 8 ∅ ∈ ω
3 elin 3167 . . . . . . . 8 (∅ ∈ (ω ∩ 𝐴) ↔ (∅ ∈ ω ∧ ∅ ∈ 𝐴))
42, 3mpbiran 882 . . . . . . 7 (∅ ∈ (ω ∩ 𝐴) ↔ ∅ ∈ 𝐴)
54biimpri 131 . . . . . 6 (∅ ∈ 𝐴 → ∅ ∈ (ω ∩ 𝐴))
6 peano2 4372 . . . . . . . . . . . 12 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
76adantr 270 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥 ∈ ω)
87a1i 9 . . . . . . . . . 10 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥 ∈ ω))
9 pm3.31 258 . . . . . . . . . 10 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥𝐴))
108, 9jcad 301 . . . . . . . . 9 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1110alimi 1385 . . . . . . . 8 (∀𝑥(𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ∀𝑥((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
12 df-ral 2358 . . . . . . . 8 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ↔ ∀𝑥(𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)))
13 elin 3167 . . . . . . . . . 10 (𝑥 ∈ (ω ∩ 𝐴) ↔ (𝑥 ∈ ω ∧ 𝑥𝐴))
14 elin 3167 . . . . . . . . . 10 (suc 𝑥 ∈ (ω ∩ 𝐴) ↔ (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴))
1513, 14imbi12i 237 . . . . . . . . 9 ((𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1615albii 1400 . . . . . . . 8 (∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ∀𝑥((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1711, 12, 163imtr4i 199 . . . . . . 7 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
18 df-ral 2358 . . . . . . 7 (∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴) ↔ ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
1917, 18sylibr 132 . . . . . 6 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))
205, 19anim12i 331 . . . . 5 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
21 omex 4370 . . . . . . 7 ω ∈ V
2221inex1 3938 . . . . . 6 (ω ∩ 𝐴) ∈ V
23 eleq2 2146 . . . . . . 7 (𝑦 = (ω ∩ 𝐴) → (∅ ∈ 𝑦 ↔ ∅ ∈ (ω ∩ 𝐴)))
24 eleq2 2146 . . . . . . . 8 (𝑦 = (ω ∩ 𝐴) → (suc 𝑥𝑦 ↔ suc 𝑥 ∈ (ω ∩ 𝐴)))
2524raleqbi1dv 2563 . . . . . . 7 (𝑦 = (ω ∩ 𝐴) → (∀𝑥𝑦 suc 𝑥𝑦 ↔ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
2623, 25anbi12d 457 . . . . . 6 (𝑦 = (ω ∩ 𝐴) → ((∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))))
2722, 26elab 2748 . . . . 5 ((ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
2820, 27sylibr 132 . . . 4 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)})
29 intss1 3677 . . . 4 ((ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} → {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ⊆ (ω ∩ 𝐴))
3028, 29syl 14 . . 3 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ⊆ (ω ∩ 𝐴))
311, 30syl5eqss 3054 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ (ω ∩ 𝐴))
32 ssid 3029 . . . 4 ω ⊆ ω
3332biantrur 297 . . 3 (ω ⊆ 𝐴 ↔ (ω ⊆ ω ∧ ω ⊆ 𝐴))
34 ssin 3206 . . 3 ((ω ⊆ ω ∧ ω ⊆ 𝐴) ↔ ω ⊆ (ω ∩ 𝐴))
3533, 34bitri 182 . 2 (ω ⊆ 𝐴 ↔ ω ⊆ (ω ∩ 𝐴))
3631, 35sylibr 132 1 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wral 2353  cin 2983  wss 2984  c0 3269   cint 3662  suc csuc 4155  ωcom 4367
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-suc 4161  df-iom 4368
This theorem is referenced by:  find  4376  finds  4377  finds2  4378  indpi  6802
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