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Theorem peano5 4246
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 4251. (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 4240 . . 3 𝜔 = {y ∣ (∅ y x y suc x y)}
2 peano1 4242 . . . . . . . 8 𝜔
3 elin 3102 . . . . . . . 8 (∅ (𝜔 ∩ A) ↔ (∅ 𝜔 A))
42, 3mpbiran 835 . . . . . . 7 (∅ (𝜔 ∩ A) ↔ ∅ A)
54biimpri 124 . . . . . 6 (∅ A → ∅ (𝜔 ∩ A))
6 peano2 4243 . . . . . . . . . . . 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 1324 . . . . . . . 8 (x(x 𝜔 → (x A → suc x A)) → x((x 𝜔 x A) → (suc x 𝜔 suc x A)))
12 df-ral 2288 . . . . . . . 8 (x 𝜔 (x A → suc x A) ↔ x(x 𝜔 → (x A → suc x A)))
13 elin 3102 . . . . . . . . . 10 (x (𝜔 ∩ A) ↔ (x 𝜔 x A))
14 elin 3102 . . . . . . . . . 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 1339 . . . . . . . 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 2288 . . . . . . 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 4241 . . . . . . 7 𝜔 V
2221inex1 3864 . . . . . 6 (𝜔 ∩ A) V
23 eleq2 2084 . . . . . . 7 (y = (𝜔 ∩ A) → (∅ y ↔ ∅ (𝜔 ∩ A)))
24 eleq2 2084 . . . . . . . 8 (y = (𝜔 ∩ A) → (suc x y ↔ suc x (𝜔 ∩ A)))
2524raleqbi1dv 2490 . . . . . . 7 (y = (𝜔 ∩ A) → (x y suc x yx (𝜔 ∩ A)suc x (𝜔 ∩ A)))
2623, 25anbi12d 445 . . . . . 6 (y = (𝜔 ∩ A) → ((∅ y x y suc x y) ↔ (∅ (𝜔 ∩ A) x (𝜔 ∩ A)suc x (𝜔 ∩ A))))
2722, 26elab 2663 . . . . 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 3603 . . . 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 2965 . 2 ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ (𝜔 ∩ A))
32 ssid 2940 . . . 4 𝜔 ⊆ 𝜔
3332biantrur 287 . . 3 (𝜔 ⊆ A ↔ (𝜔 ⊆ 𝜔 𝜔 ⊆ A))
34 ssin 3135 . . 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 1226   = wceq 1228   wcel 1375  {cab 2009  wral 2283  cin 2892  wss 2893  c0 3200   cint 3588  suc csuc 4049  𝜔com 4238
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-nul 3856  ax-pow 3900  ax-pr 3917  ax-un 4118  ax-iinf 4236
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-int 3589  df-suc 4055  df-iom 4239
This theorem is referenced by:  find  4247  finds  4248  finds2  4249
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