ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  peano5 GIF version

Theorem peano5 4582
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 4587. (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 4576 . . 3 ω = {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}
2 peano1 4578 . . . . . . . 8 ∅ ∈ ω
3 elin 3310 . . . . . . . 8 (∅ ∈ (ω ∩ 𝐴) ↔ (∅ ∈ ω ∧ ∅ ∈ 𝐴))
42, 3mpbiran 935 . . . . . . 7 (∅ ∈ (ω ∩ 𝐴) ↔ ∅ ∈ 𝐴)
54biimpri 132 . . . . . 6 (∅ ∈ 𝐴 → ∅ ∈ (ω ∩ 𝐴))
6 peano2 4579 . . . . . . . . . . . 12 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
76adantr 274 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥 ∈ ω)
87a1i 9 . . . . . . . . . 10 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥 ∈ ω))
9 pm3.31 260 . . . . . . . . . 10 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥𝐴))
108, 9jcad 305 . . . . . . . . 9 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1110alimi 1448 . . . . . . . 8 (∀𝑥(𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ∀𝑥((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
12 df-ral 2453 . . . . . . . 8 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ↔ ∀𝑥(𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)))
13 elin 3310 . . . . . . . . . 10 (𝑥 ∈ (ω ∩ 𝐴) ↔ (𝑥 ∈ ω ∧ 𝑥𝐴))
14 elin 3310 . . . . . . . . . 10 (suc 𝑥 ∈ (ω ∩ 𝐴) ↔ (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴))
1513, 14imbi12i 238 . . . . . . . . 9 ((𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1615albii 1463 . . . . . . . 8 (∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ∀𝑥((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1711, 12, 163imtr4i 200 . . . . . . 7 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
18 df-ral 2453 . . . . . . 7 (∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴) ↔ ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
1917, 18sylibr 133 . . . . . 6 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))
205, 19anim12i 336 . . . . 5 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
21 omex 4577 . . . . . . 7 ω ∈ V
2221inex1 4123 . . . . . 6 (ω ∩ 𝐴) ∈ V
23 eleq2 2234 . . . . . . 7 (𝑦 = (ω ∩ 𝐴) → (∅ ∈ 𝑦 ↔ ∅ ∈ (ω ∩ 𝐴)))
24 eleq2 2234 . . . . . . . 8 (𝑦 = (ω ∩ 𝐴) → (suc 𝑥𝑦 ↔ suc 𝑥 ∈ (ω ∩ 𝐴)))
2524raleqbi1dv 2673 . . . . . . 7 (𝑦 = (ω ∩ 𝐴) → (∀𝑥𝑦 suc 𝑥𝑦 ↔ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
2623, 25anbi12d 470 . . . . . 6 (𝑦 = (ω ∩ 𝐴) → ((∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))))
2722, 26elab 2874 . . . . 5 ((ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
2820, 27sylibr 133 . . . 4 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)})
29 intss1 3846 . . . 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, 30eqsstrid 3193 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ (ω ∩ 𝐴))
32 ssid 3167 . . . 4 ω ⊆ ω
3332biantrur 301 . . 3 (ω ⊆ 𝐴 ↔ (ω ⊆ ω ∧ ω ⊆ 𝐴))
34 ssin 3349 . . 3 ((ω ⊆ ω ∧ ω ⊆ 𝐴) ↔ ω ⊆ (ω ∩ 𝐴))
3533, 34bitri 183 . 2 (ω ⊆ 𝐴 ↔ ω ⊆ (ω ∩ 𝐴))
3631, 35sylibr 133 1 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1346   = wceq 1348  wcel 2141  {cab 2156  wral 2448  cin 3120  wss 3121  c0 3414   cint 3831  suc csuc 4350  ωcom 4574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575
This theorem is referenced by:  find  4583  finds  4584  finds2  4585  indpi  7304
  Copyright terms: Public domain W3C validator