Theorem find 3118

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:   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 788	. 2 |- A (_ om
3		ax-1 4	. . . . . . . . 9 |- (suc x e. A -> (x e. om -> suc x e. A))
4	3	r19.20si 1682	. . . . . . . 8 |- (A.x e. A suc x e. A -> A.x e. A (x e. om -> suc x e. A))
5		ralcom3 1753	. . . . . . . 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 776	. . . . 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 776	. . . . 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 3116	. . . 4 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)
14	13	3adant1 794	. . 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 2049	1 |- A = om

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 772   = wceq 1099   e. wcel 1105  A.wral 1621   (_ wss 2018  (/)c0 2251  suc csuc 2913  omcom 3094

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095

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