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Theorem find 7834
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.) (Proof shortened by Wolf Lammen, 28-May-2024.)
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 1140 . 2 𝐴 ⊆ ω
3 3simpc 1151 . . . 4 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
4 df-ral 3062 . . . . . 6 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
5 alral 3075 . . . . . 6 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
64, 5sylbi 216 . . . . 5 (∀𝑥𝐴 suc 𝑥𝐴 → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
76anim2i 618 . . . 4 ((∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
81, 3, 7mp2b 10 . . 3 (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
9 peano5 7831 . . 3 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
108, 9ax-mp 5 . 2 ω ⊆ 𝐴
112, 10eqssi 3961 1 𝐴 = ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  wral 3061  wss 3911  c0 4283  suc csuc 6320  ωcom 7803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-om 7804
This theorem is referenced by: (None)
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