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Theorem find 6983
Description: The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that 𝐴 is a set of natural numbers, zero belongs to 𝐴, and given any member of 𝐴 the member's successor also belongs to 𝐴. The conclusion is that every natural number is in 𝐴. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
find.1 (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)
Assertion
Ref Expression
find 𝐴 = ω
Distinct variable group:   𝑥,𝐴

Proof of Theorem find
StepHypRef Expression
1 find.1 . . 3 (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)
21simp1i 1063 . 2 𝐴 ⊆ ω
3 3simpc 1053 . . . . 5 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
41, 3ax-mp 5 . . . 4 (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)
5 df-ral 2901 . . . . . 6 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
6 alral 2912 . . . . . 6 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
75, 6sylbi 206 . . . . 5 (∀𝑥𝐴 suc 𝑥𝐴 → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
87anim2i 591 . . . 4 ((∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
94, 8ax-mp 5 . . 3 (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
10 peano5 6981 . . 3 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
119, 10ax-mp 5 . 2 ω ⊆ 𝐴
122, 11eqssi 3584 1 𝐴 = ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wral 2896  wss 3540  c0 3874  suc csuc 5642  ωcom 6957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958
This theorem is referenced by: (None)
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