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Theorem find 6864
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 1062 . 2 𝐴 ⊆ ω
3 3simpc 1052 . . . . 5 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
41, 3ax-mp 5 . . . 4 (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)
5 df-ral 2805 . . . . . 6 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
6 alral 2816 . . . . . 6 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
75, 6sylbi 205 . . . . 5 (∀𝑥𝐴 suc 𝑥𝐴 → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
87anim2i 590 . . . 4 ((∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
94, 8ax-mp 5 . . 3 (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
10 peano5 6862 . . 3 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
119, 10ax-mp 5 . 2 ω ⊆ 𝐴
122, 11eqssi 3488 1 𝐴 = ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wal 1472   = wceq 1474  wcel 1938  wral 2800  wss 3444  c0 3777  suc csuc 5532  ωcom 6838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6728
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-tr 4579  df-eprel 4843  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-om 6839
This theorem is referenced by: (None)
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