Theorem finds 3146

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 2701	. . . . . 6 |- (/) e. V
3		finds.1	. . . . . 6 |- (x = (/) -> (ph <-> ps))
4	2, 3	elab 1888	. . . . 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 1804	. . . . . . 7 |- y e. V
8		finds.2	. . . . . . 7 |- (x = y -> (ph <-> ch))
9	7, 8	elab 1888	. . . . . 6 |- (y e. {x | ph} <-> ch)
10	7	sucex 3040	. . . . . . 7 |- suc y e. V
11		finds.3	. . . . . . 7 |- (x = suc y -> (ph <-> th))
12	10, 11	elab 1888	. . . . . 6 |- (suc y e. {x | ph} <-> th)
13	6, 9, 12	3imtr4g 551	. . . . 5 |- (y e. om -> (y e. {x | ph} -> suc y e. {x | ph}))
14	13	rgen 1690	. . . 4 |- A.y e. om (y e. {x | ph} -> suc y e. {x | ph})
15		peano5 3143	. . . 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 695	. . 3 |- om (_ {x | ph}
17	16	sseli 2055	. 2 |- (A e. om -> A e. {x | ph})
18		finds.4	. . 3 |- (x = A -> (ph <-> ta))
19	18	elabg 1890	. 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 953   e. wcel 955  {cab 1456  A.wral 1637   (_ wss 2037  (/)c0 2270  suc csuc 2940  omcom 3121

This theorem is referenced by:  findsg 3147  findes 3150  nnacl 4213  nnmcl 4214  nnecl 4215  nnacom 4217  nnmsucr 4224  nnmcom 4225  nneob 4239  nneneq 4492  pssnn 4513  inf3lem1 4585  inf3lem2 4586  om2uzuz 6234  om2uzlt 6235  findfvcl 10323

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122

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