Theorem find 3150

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.

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	3simp1i 790	. 2 ⊢ A ⊆ ω
3		ax-1 4	. . . . . . . . 9 ⊢ (suc x ∈ A → (x ∈ ω → suc x ∈ A))
4	3	r19.20si 1703	. . . . . . . 8 ⊢ (∀x ∈ A suc x ∈ A → ∀x ∈ A (x ∈ ω → suc x ∈ A))
5		ralcom3 1774	. . . . . . . 8 ⊢ (∀x ∈ A (x ∈ ω → suc x ∈ A) ↔ ∀x ∈ ω (x ∈ A → suc x ∈ A))
6	4, 5	sylib 198	. . . . . . 7 ⊢ (∀x ∈ A suc x ∈ A → ∀x ∈ ω (x ∈ A → suc x ∈ A))
7	6	anim2i 335	. . . . . 6 ⊢ ((∅ ∈ A ⋀ ∀x ∈ A suc x ∈ A) → (∅ ∈ A ⋀ ∀x ∈ ω (x ∈ A → suc x ∈ A)))
8	7	anim2i 335	. . . . 5 ⊢ ((A ⊆ ω ⋀ (∅ ∈ A ⋀ ∀x ∈ A suc x ∈ A)) → (A ⊆ ω ⋀ (∅ ∈ A ⋀ ∀x ∈ ω (x ∈ A → suc x ∈ A))))
9		3anass 778	. . . . 5 ⊢ ((A ⊆ ω ⋀ ∅ ∈ A ⋀ ∀x ∈ A suc x ∈ A) ↔ (A ⊆ ω ⋀ (∅ ∈ A ⋀ ∀x ∈ A suc x ∈ A)))
10		3anass 778	. . . . 5 ⊢ ((A ⊆ ω ⋀ ∅ ∈ A ⋀ ∀x ∈ ω (x ∈ A → suc x ∈ A)) ↔ (A ⊆ ω ⋀ (∅ ∈ A ⋀ ∀x ∈ ω (x ∈ A → suc x ∈ A))))
11	8, 9, 10	3imtr4 219	. . . 4 ⊢ ((A ⊆ ω ⋀ ∅ ∈ A ⋀ ∀x ∈ A suc x ∈ A) → (A ⊆ ω ⋀ ∅ ∈ A ⋀ ∀x ∈ ω (x ∈ A → suc x ∈ A)))
12	1, 11	ax-mp 7	. . 3 ⊢ (A ⊆ ω ⋀ ∅ ∈ A ⋀ ∀x ∈ ω (x ∈ A → suc x ∈ A))
13		peano5 3148	. . . 4 ⊢ ((∅ ∈ A ⋀ ∀x ∈ ω (x ∈ A → suc x ∈ A)) → ω ⊆ A)
14	13	3adant1 796	. . 3 ⊢ ((A ⊆ ω ⋀ ∅ ∈ A ⋀ ∀x ∈ ω (x ∈ A → suc x ∈ A)) → ω ⊆ A)
15	12, 14	ax-mp 7	. 2 ⊢ ω ⊆ A
16	2, 15	eqssi 2074	1 ⊢ A = ω

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 774   = wceq 954   ∈ wcel 956  ∀wral 1642   ⊆ wss 2043  ∅c0 2276  suc csuc 2945  ωcom 3126

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127

Copyright terms: Public domain