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

Assertion

Ref	Expression
find	|- A = om

Distinct variable group(s):   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		3simp1 594	. . 3 |- ((A (_ om /\ (/) e. A /\ A.x e. A suc x e. A) -> A (_ om)
3	1, 2	ax-mp 6	. 2 |- A (_ om
4		ax-1 3	. . . . . . . . 9 |- (suc x e. A -> (x e. om -> suc x e. A))
5	4	r19.20si 1254	. . . . . . . 8 |- (A.x e. A suc x e. A -> A.x e. A (x e. om -> suc x e. A))
6		ralcom3 1315	. . . . . . . 8 |- (A.x e. A (x e. om -> suc x e. A) <-> A.x e. om (x e. A -> suc x e. A))
7	5, 6	sylib 173	. . . . . . 7 |- (A.x e. A suc x e. A -> A.x e. om (x e. A -> suc x e. A))
8	7	anim2i 270	. . . . . 6 |- (((/) e. A /\ A.x e. A suc x e. A) -> ((/) e. A /\ A.x e. om (x e. A -> suc x e. A)))
9	8	anim2i 270	. . . . 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))))
10		3anass 585	. . . . 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)))
11		3anass 585	. . . . 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))))
12	9, 10, 11	3imtr4 192	. . . 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)))
13	1, 12	ax-mp 6	. . 3 |- (A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A))
14		peano5 2394	. . . 4 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)
15	14	3adant1 597	. . 3 |- ((A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)
16	13, 15	ax-mp 6	. 2 |- om (_ A
17	3, 16	eqssi 1517	1 |- A = om

Syntax hints:   -> wi 2   /\ wa 196   /\ w3a 581   = wceq 1091   e. wcel 1092  A.wral 1201   (_ wss 1487  (/)c0 1707  suc csuc 2201  omcom 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

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