Theorem find 2396

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-01. 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.

Hypothesis

Ref	Expression
find.1	⊢ (A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)

Assertion

Ref	Expression
find	⊢ A = ω

Distinct variable group(s):   x,A

Proof of Theorem find

Step	Hyp	Ref	Expression
1		find.1	. . 3 ⊢ (A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)
2		3simp1 594	. . 3 ⊢ ((A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → A ⊆ ω)
3	1, 2	ax-mp 6	. 2 ⊢ A ⊆ ω
4		ax-1 3	. . . . . . . . 9 ⊢ (suc x ∈ A → (x ∈ ω → suc x ∈ A))
5	4	r19.20si 1254	. . . . . . . 8 ⊢ (∀x ∈ A suc x ∈ A → ∀x ∈ A (x ∈ ω → suc x ∈ A))
6		ralcom3 1315	. . . . . . . 8 ⊢ (∀x ∈ A (x ∈ ω → suc x ∈ A) ↔ ∀x ∈ ω (x ∈ A → suc x ∈ A))
7	5, 6	sylib 173	. . . . . . 7 ⊢ (∀x ∈ A suc x ∈ A → ∀x ∈ ω (x ∈ A → suc x ∈ A))
8	7	anim2i 270	. . . . . 6 ⊢ ((∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → (∅ ∈ A ∧ ∀x ∈ ω (x ∈ A → suc x ∈ A)))
9	8	anim2i 270	. . . . 5 ⊢ ((A ⊆ ω ∧ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) → (A ⊆ ω ∧ (∅ ∈ A ∧ ∀x ∈ ω (x ∈ A → suc x ∈ A))))
10		3anass 585	. . . . 5 ⊢ ((A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) ↔ (A ⊆ ω ∧ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)))
11		3anass 585	. . . . 5 ⊢ ((A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ ω (x ∈ A → suc x ∈ A)) ↔ (A ⊆ ω ∧ (∅ ∈ A ∧ ∀x ∈ ω (x ∈ A → suc x ∈ A))))
12	9, 10, 11	3imtr4 192	. . . 4 ⊢ ((A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → (A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ ω (x ∈ A → suc x ∈ A)))
13	1, 12	ax-mp 6	. . 3 ⊢ (A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ ω (x ∈ A → suc x ∈ A))
14		peano5 2394	. . . 4 ⊢ ((∅ ∈ A ∧ ∀x ∈ ω (x ∈ A → suc x ∈ A)) → ω ⊆ A)
15	14	3adant1 597	. . 3 ⊢ ((A ⊆ ω ∧ ∅ ∈ A ∧ ∀x ∈ ω (x ∈ A → suc x ∈ A)) → ω ⊆ A)
16	13, 15	ax-mp 6	. 2 ⊢ ω ⊆ A
17	3, 16	eqssi 1517	1 ⊢ A = ω

Syntax hints:   → wi 2   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  ∅c0 1707  suc csuc 2201  ωcom 2372

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373

metamath.org