Theorem find 3155

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

Assertion

Ref	Expression
find	|- A = om

Distinct variable group:   x,A

Proof of Theorem find

Step	Hyp	Ref	Expression
1		find.1	. . 3 |- (A (_ om /\ (/) e. A /\ A.x e. A suc x e. A)
2	1	3simp1i 791	. 2 |- A (_ om
3		ax-1 4	. . . . . . . . 9 |- (suc x e. A -> (x e. om -> suc x e. A))
4	3	r19.20si 1706	. . . . . . . 8 |- (A.x e. A suc x e. A -> A.x e. A (x e. om -> suc x e. A))
5		ralcom3 1777	. . . . . . . 8 |- (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 198	. . . . . . 7 |- (A.x e. A suc x e. A -> A.x e. om (x e. A -> suc x e. A))
7	6	anim2i 335	. . . . . 6 |- (((/) e. A /\ A.x e. A suc x e. A) -> ((/) e. A /\ A.x e. om (x e. A -> suc x e. A)))
8	7	anim2i 335	. . . . 5 |- ((A (_ om /\ ((/) e. A /\ A.x e. A suc x e. A)) -> (A (_ om /\ ((/) e. A /\ A.x e. om (x e. A -> suc x e. A))))
9		3anass 779	. . . . 5 |- ((A (_ om /\ (/) e. A /\ A.x e. A suc x e. A) <-> (A (_ om /\ ((/) e. A /\ A.x e. A suc x e. A)))
10		3anass 779	. . . . 5 |- ((A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) <-> (A (_ om /\ ((/) e. A /\ A.x e. om (x e. A -> suc x e. A))))
11	8, 9, 10	3imtr4 219	. . . 4 |- ((A (_ om /\ (/) e. A /\ A.x e. A suc x e. A) -> (A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)))
12	1, 11	ax-mp 7	. . 3 |- (A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A))
13		peano5 3153	. . . 4 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)
14	13	3adant1 797	. . 3 |- ((A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)
15	12, 14	ax-mp 7	. 2 |- om (_ A
16	2, 15	eqssi 2078	1 |- A = om

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  (/)c0 2280  suc csuc 2950  omcom 3131

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132

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