Theorem find 7914

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 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
find	⊢ 𝐴 = ω

Distinct variable group:   𝑥,𝐴

Proof of Theorem find

Step	Hyp	Ref	Expression
1		find.1	. . 3 ⊢ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)
2	1	simp1i 1157	. 2 ⊢ 𝐴 ⊆ ω
3		3simpc 1168	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
4		df-ral 3080	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
5		alral 3094	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
6	4, 5	sylbi 220	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
7	6	anim2i 629	. . . 4 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
8	1, 3, 7	mp2b 10	. . 3 ⊢ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
9		peano5 7912	. . 3 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
10	8, 9	ax-mp 5	. 2 ⊢ ω ⊆ 𝐴
11	2, 10	eqssi 3928	1 ⊢ 𝐴 = ω

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2146  ∀wral 3079   ⊆ wss 3899  ∅c0 4280  suc csuc 6370  ωcom 7884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-tr 5215  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-om 7885

This theorem is used by: (None)