Theorem finds 3152

Description: Principle of Finite Induction (inference schema) with implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136.

Hypotheses

Ref	Expression
finds.1	|- (x = (/) -> (ph <-> ps))
finds.2	|- (x = y -> (ph <-> ch))
finds.3	|- (x = suc y -> (ph <-> th))
finds.4	|- (x = A -> (ph <-> ta))
finds.5	|- ps
finds.6	|- (y e. om -> (ch -> th))

Assertion

Ref	Expression
finds	|- (A e. om -> ta)

Distinct variable groups:   x,y   x,A   ps,x   ch,x   th,x   ta,x   ph,y

Proof of Theorem finds

Step	Hyp	Ref	Expression
1		finds.5	. . . . 5 |- ps
2		0ex 2707	. . . . . 6 |- (/) e. V
3		finds.1	. . . . . 6 |- (x = (/) -> (ph <-> ps))
4	2, 3	elab 1894	. . . . 5 |- ((/) e. {x | ph} <-> ps)
5	1, 4	mpbir 190	. . . 4 |- (/) e. {x | ph}
6		finds.6	. . . . . 6 |- (y e. om -> (ch -> th))
7		visset 1810	. . . . . . 7 |- y e. V
8		finds.2	. . . . . . 7 |- (x = y -> (ph <-> ch))
9	7, 8	elab 1894	. . . . . 6 |- (y e. {x | ph} <-> ch)
10	7	sucex 3046	. . . . . . 7 |- suc y e. V
11		finds.3	. . . . . . 7 |- (x = suc y -> (ph <-> th))
12	10, 11	elab 1894	. . . . . 6 |- (suc y e. {x | ph} <-> th)
13	6, 9, 12	3imtr4g 552	. . . . 5 |- (y e. om -> (y e. {x | ph} -> suc y e. {x | ph}))
14	13	rgen 1696	. . . 4 |- A.y e. om (y e. {x | ph} -> suc y e. {x | ph})
15		peano5 3149	. . . 4 |- (((/) e. {x | ph} /\ A.y e. om (y e. {x | ph} -> suc y e. {x | ph})) -> om (_ {x | ph})
16	5, 14, 15	mp2an 696	. . 3 |- om (_ {x | ph}
17	16	sseli 2062	. 2 |- (A e. om -> A e. {x | ph})
18		finds.4	. . 3 |- (x = A -> (ph <-> ta))
19	18	elabg 1896	. 2 |- (A e. om -> (A e. {x | ph} <-> ta))
20	17, 19	mpbid 195	1 |- (A e. om -> ta)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  {cab 1462  A.wral 1643   (_ wss 2044  (/)c0 2277  suc csuc 2946  omcom 3127

This theorem is referenced by:  findsg 3153  findes 3156  nnacl 4222  nnmcl 4223  nnecl 4224  nnacom 4226  nnmsucr 4233  nnmcom 4234  nneob 4248  nneneq 4501  pssnn 4522  inf3lem1 4596  inf3lem2 4597  om2uzuz 6247  om2uzlt 6248  findfvcl 10372

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128

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