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Theorem peano5 4299
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 4304. (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 4293 . . 3 ω = {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}
2 peano1 4295 . . . . . . . 8 ∅ ∈ ω
3 elin 3123 . . . . . . . 8 (∅ ∈ (ω ∩ 𝐴) ↔ (∅ ∈ ω ∧ ∅ ∈ 𝐴))
42, 3mpbiran 847 . . . . . . 7 (∅ ∈ (ω ∩ 𝐴) ↔ ∅ ∈ 𝐴)
54biimpri 124 . . . . . 6 (∅ ∈ 𝐴 → ∅ ∈ (ω ∩ 𝐴))
6 peano2 4296 . . . . . . . . . . . 12 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
76adantr 261 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥 ∈ ω)
87a1i 9 . . . . . . . . . 10 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥 ∈ ω))
9 pm3.31 249 . . . . . . . . . 10 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥𝐴))
108, 9jcad 291 . . . . . . . . 9 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1110alimi 1344 . . . . . . . 8 (∀𝑥(𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ∀𝑥((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
12 df-ral 2308 . . . . . . . 8 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ↔ ∀𝑥(𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)))
13 elin 3123 . . . . . . . . . 10 (𝑥 ∈ (ω ∩ 𝐴) ↔ (𝑥 ∈ ω ∧ 𝑥𝐴))
14 elin 3123 . . . . . . . . . 10 (suc 𝑥 ∈ (ω ∩ 𝐴) ↔ (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴))
1513, 14imbi12i 228 . . . . . . . . 9 ((𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1615albii 1359 . . . . . . . 8 (∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ∀𝑥((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1711, 12, 163imtr4i 190 . . . . . . 7 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
18 df-ral 2308 . . . . . . 7 (∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴) ↔ ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
1917, 18sylibr 137 . . . . . 6 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))
205, 19anim12i 321 . . . . 5 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
21 omex 4294 . . . . . . 7 ω ∈ V
2221inex1 3888 . . . . . 6 (ω ∩ 𝐴) ∈ V
23 eleq2 2101 . . . . . . 7 (𝑦 = (ω ∩ 𝐴) → (∅ ∈ 𝑦 ↔ ∅ ∈ (ω ∩ 𝐴)))
24 eleq2 2101 . . . . . . . 8 (𝑦 = (ω ∩ 𝐴) → (suc 𝑥𝑦 ↔ suc 𝑥 ∈ (ω ∩ 𝐴)))
2524raleqbi1dv 2510 . . . . . . 7 (𝑦 = (ω ∩ 𝐴) → (∀𝑥𝑦 suc 𝑥𝑦 ↔ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
2623, 25anbi12d 442 . . . . . 6 (𝑦 = (ω ∩ 𝐴) → ((∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))))
2722, 26elab 2684 . . . . 5 ((ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
2820, 27sylibr 137 . . . 4 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)})
29 intss1 3627 . . . 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 2986 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ (ω ∩ 𝐴))
32 ssid 2961 . . . 4 ω ⊆ ω
3332biantrur 287 . . 3 (ω ⊆ 𝐴 ↔ (ω ⊆ ω ∧ ω ⊆ 𝐴))
34 ssin 3156 . . 3 ((ω ⊆ ω ∧ ω ⊆ 𝐴) ↔ ω ⊆ (ω ∩ 𝐴))
3533, 34bitri 173 . 2 (ω ⊆ 𝐴 ↔ ω ⊆ (ω ∩ 𝐴))
3631, 35sylibr 137 1 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wal 1241   = wceq 1243  wcel 1393  {cab 2026  wral 2303  cin 2913  wss 2914  c0 3221   cint 3612  suc csuc 4089  ωcom 4291
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-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4157  ax-iinf 4289
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-uni 3578  df-int 3613  df-suc 4095  df-iom 4292
This theorem is referenced by:  find  4300  finds  4301  finds2  4302  indpi  6412
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