Theorem findOLD 3978

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 C_ om /\ (/) e. A /\ A.x e. A suc x e. A)

Assertion

Ref	Expression
findOLD	|- A = om

Distinct variable group:   x,A

Proof of Theorem findOLD

Step	Hyp	Ref	Expression
1		find.1	. . 3 |- (A C_ om /\ (/) e. A /\ A.x e. A suc x e. A)
2	1	simp1i 885	. 2 |- A C_ om
3		ax-1 4	. . . . . . 7 |- (suc x e. A -> (x e. om -> suc x e. A))
4	3	ralimi 2168	. . . . . 6 |- (A.x e. A suc x e. A -> A.x e. A (x e. om -> suc x e. A))
5		ralcom3 2246	. . . . . 6 |- (A.x e. A (x e. om -> suc x e. A) <-> A.x e. om (x e. A -> suc x e. A))
6	4, 5	sylib 215	. . . . 5 |- (A.x e. A suc x e. A -> A.x e. om (x e. A -> suc x e. A))
7	6	3anim3i 1055	. . . 4 |- ((A C_ om /\ (/) e. A /\ A.x e. A suc x e. A) -> (A C_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)))
8	1, 7	ax-mp 7	. . 3 |- (A C_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A))
9		peano5 3975	. . . 4 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om C_ A)
10	9	3adant1 894	. . 3 |- ((A C_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om C_ A)
11	8, 10	ax-mp 7	. 2 |- om C_ A
12	2, 11	eqssi 2632	1 |- A = om

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  (/)c0 2875  suc csuc 3659  omcom 3949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950

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