Theorem finds 3979

Description: Principle of Finite Induction (inference schema), using 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 3446	. . . . . 6 |- (/) e. _V
3		finds.1	. . . . . 6 |- (x = (/) -> (ph <-> ps))
4	2, 3	elab 2403	. . . . 5 |- ((/) e. {x | ph} <-> ps)
5	1, 4	mpbir 207	. . . 4 |- (/) e. {x | ph}
6		finds.6	. . . . . 6 |- (y e. om -> (ch -> th))
7		visset 2295	. . . . . . 7 |- y e. _V
8		finds.2	. . . . . . 7 |- (x = y -> (ph <-> ch))
9	7, 8	elab 2403	. . . . . 6 |- (y e. {x | ph} <-> ch)
10	7	sucex 3892	. . . . . . 7 |- suc y e. _V
11		finds.3	. . . . . . 7 |- (x = suc y -> (ph <-> th))
12	10, 11	elab 2403	. . . . . 6 |- (suc y e. {x | ph} <-> th)
13	6, 9, 12	3imtr4g 612	. . . . 5 |- (y e. om -> (y e. {x | ph} -> suc y e. {x | ph}))
14	13	rgen 2159	. . . 4 |- A.y e. om (y e. {x | ph} -> suc y e. {x | ph})
15		peano5 3975	. . . 4 |- (((/) e. {x | ph} /\ A.y e. om (y e. {x | ph} -> suc y e. {x | ph})) -> om C_ {x | ph})
16	5, 14, 15	mp2an 761	. . 3 |- om C_ {x | ph}
17	16	sseli 2617	. 2 |- (A e. om -> A e. {x | ph})
18		finds.4	. . 3 |- (x = A -> (ph <-> ta))
19	18	elabg 2405	. 2 |- (A e. om -> (A e. {x | ph} <-> ta))
20	17, 19	mpbid 212	1 |- (A e. om -> ta)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105   C_ wss 2593  (/)c0 2875  suc csuc 3659  omcom 3949

This theorem is referenced by:  findsg 3980  findes 3983  nnaclOLD 5282  nnmclOLD 5284  nneclOLD 5286  nnacom 5288  nnmsucrOLD 5296  nnmcomOLD 5298  nneob 5312  nneneq 5606  pssnn 5628  inf3lem1 5719  inf3lem2 5720  om2uzuzi 7708  om2uzlti 7709  indexfi 10174  findcardOLD 10179  fbssint 10279  findfvcl 14253  fictb 15371  finsschain 15373  neibastop2lem1 15519  fcluscomplem 15620  indexfiOLD 15755

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950

Copyright terms: Public domain