Theorem find 7252

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 ∧ ∀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 1134	. 2 ⊢ 𝐴 ⊆ ω
3		3simpc 1147	. . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
4	1, 3	ax-mp 5	. . . 4 ⊢ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)
5		df-ral 3051	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
6		alral 3062	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
7	5, 6	sylbi 207	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
8	7	anim2i 594	. . . 4 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
9	4, 8	ax-mp 5	. . 3 ⊢ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
10		peano5 7250	. . 3 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
11	9, 10	ax-mp 5	. 2 ⊢ ω ⊆ 𝐴
12	2, 11	eqssi 3756	1 ⊢ 𝐴 = ω

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072  ∀wal 1626   = wceq 1628   ∈ wcel 2135  ∀wral 3046   ⊆ wss 3711  ∅c0 4054  suc csuc 5882  ωcom 7226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051  ax-un 7110

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-tr 4901  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-om 7227

This theorem is referenced by: (None)