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Theorem peano5 4263
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 4268. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
peano5 ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
Distinct variable group:   x,A

Proof of Theorem peano5
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfom3 4257 . . 3 𝜔 = {y ∣ (∅ y x y suc x y)}
2 peano1 4259 . . . . . . . 8 𝜔
3 elin 3120 . . . . . . . 8 (∅ (𝜔 ∩ A) ↔ (∅ 𝜔 A))
42, 3mpbiran 846 . . . . . . 7 (∅ (𝜔 ∩ A) ↔ ∅ A)
54biimpri 124 . . . . . 6 (∅ A → ∅ (𝜔 ∩ A))
6 peano2 4260 . . . . . . . . . . . 12 (x 𝜔 → suc x 𝜔)
76adantr 261 . . . . . . . . . . 11 ((x 𝜔 x A) → suc x 𝜔)
87a1i 9 . . . . . . . . . 10 ((x 𝜔 → (x A → suc x A)) → ((x 𝜔 x A) → suc x 𝜔))
9 pm3.31 249 . . . . . . . . . 10 ((x 𝜔 → (x A → suc x A)) → ((x 𝜔 x A) → suc x A))
108, 9jcad 291 . . . . . . . . 9 ((x 𝜔 → (x A → suc x A)) → ((x 𝜔 x A) → (suc x 𝜔 suc x A)))
1110alimi 1341 . . . . . . . 8 (x(x 𝜔 → (x A → suc x A)) → x((x 𝜔 x A) → (suc x 𝜔 suc x A)))
12 df-ral 2305 . . . . . . . 8 (x 𝜔 (x A → suc x A) ↔ x(x 𝜔 → (x A → suc x A)))
13 elin 3120 . . . . . . . . . 10 (x (𝜔 ∩ A) ↔ (x 𝜔 x A))
14 elin 3120 . . . . . . . . . 10 (suc x (𝜔 ∩ A) ↔ (suc x 𝜔 suc x A))
1513, 14imbi12i 228 . . . . . . . . 9 ((x (𝜔 ∩ A) → suc x (𝜔 ∩ A)) ↔ ((x 𝜔 x A) → (suc x 𝜔 suc x A)))
1615albii 1356 . . . . . . . 8 (x(x (𝜔 ∩ A) → suc x (𝜔 ∩ A)) ↔ x((x 𝜔 x A) → (suc x 𝜔 suc x A)))
1711, 12, 163imtr4i 190 . . . . . . 7 (x 𝜔 (x A → suc x A) → x(x (𝜔 ∩ A) → suc x (𝜔 ∩ A)))
18 df-ral 2305 . . . . . . 7 (x (𝜔 ∩ A)suc x (𝜔 ∩ A) ↔ x(x (𝜔 ∩ A) → suc x (𝜔 ∩ A)))
1917, 18sylibr 137 . . . . . 6 (x 𝜔 (x A → suc x A) → x (𝜔 ∩ A)suc x (𝜔 ∩ A))
205, 19anim12i 321 . . . . 5 ((∅ A x 𝜔 (x A → suc x A)) → (∅ (𝜔 ∩ A) x (𝜔 ∩ A)suc x (𝜔 ∩ A)))
21 omex 4258 . . . . . . 7 𝜔 V
2221inex1 3881 . . . . . 6 (𝜔 ∩ A) V
23 eleq2 2098 . . . . . . 7 (y = (𝜔 ∩ A) → (∅ y ↔ ∅ (𝜔 ∩ A)))
24 eleq2 2098 . . . . . . . 8 (y = (𝜔 ∩ A) → (suc x y ↔ suc x (𝜔 ∩ A)))
2524raleqbi1dv 2507 . . . . . . 7 (y = (𝜔 ∩ A) → (x y suc x yx (𝜔 ∩ A)suc x (𝜔 ∩ A)))
2623, 25anbi12d 442 . . . . . 6 (y = (𝜔 ∩ A) → ((∅ y x y suc x y) ↔ (∅ (𝜔 ∩ A) x (𝜔 ∩ A)suc x (𝜔 ∩ A))))
2722, 26elab 2681 . . . . 5 ((𝜔 ∩ A) {y ∣ (∅ y x y suc x y)} ↔ (∅ (𝜔 ∩ A) x (𝜔 ∩ A)suc x (𝜔 ∩ A)))
2820, 27sylibr 137 . . . 4 ((∅ A x 𝜔 (x A → suc x A)) → (𝜔 ∩ A) {y ∣ (∅ y x y suc x y)})
29 intss1 3620 . . . 4 ((𝜔 ∩ A) {y ∣ (∅ y x y suc x y)} → {y ∣ (∅ y x y suc x y)} ⊆ (𝜔 ∩ A))
3028, 29syl 14 . . 3 ((∅ A x 𝜔 (x A → suc x A)) → {y ∣ (∅ y x y suc x y)} ⊆ (𝜔 ∩ A))
311, 30syl5eqss 2983 . 2 ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ (𝜔 ∩ A))
32 ssid 2958 . . . 4 𝜔 ⊆ 𝜔
3332biantrur 287 . . 3 (𝜔 ⊆ A ↔ (𝜔 ⊆ 𝜔 𝜔 ⊆ A))
34 ssin 3153 . . 3 ((𝜔 ⊆ 𝜔 𝜔 ⊆ A) ↔ 𝜔 ⊆ (𝜔 ∩ A))
3533, 34bitri 173 . 2 (𝜔 ⊆ A ↔ 𝜔 ⊆ (𝜔 ∩ A))
3631, 35sylibr 137 1 ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wral 2300  cin 2910  wss 2911  c0 3218   cint 3605  suc csuc 4067  𝜔com 4255
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-iinf 4253
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-uni 3571  df-int 3606  df-suc 4073  df-iom 4256
This theorem is referenced by:  find  4264  finds  4265  finds2  4266  indpi  6319
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