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Theorem finds 4698
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. (Contributed by NM, 14-Apr-1995.)
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
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)    ta( y)    A( y)

Proof of Theorem finds
StepHypRef Expression
1 finds.5 . . . . 5  |-  ps
2 0ex 4166 . . . . . 6  |-  (/)  e.  _V
3 finds.1 . . . . . 6  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
42, 3elab 2927 . . . . 5  |-  ( (/)  e.  { x  |  ph } 
<->  ps )
51, 4mpbir 200 . . . 4  |-  (/)  e.  {
x  |  ph }
6 finds.6 . . . . . 6  |-  ( y  e.  om  ->  ( ch  ->  th ) )
7 vex 2804 . . . . . . 7  |-  y  e. 
_V
8 finds.2 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
97, 8elab 2927 . . . . . 6  |-  ( y  e.  { x  | 
ph }  <->  ch )
107sucex 4618 . . . . . . 7  |-  suc  y  e.  _V
11 finds.3 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
1210, 11elab 2927 . . . . . 6  |-  ( suc  y  e.  { x  |  ph }  <->  th )
136, 9, 123imtr4g 261 . . . . 5  |-  ( y  e.  om  ->  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )
1413rgen 2621 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ph }  ->  suc  y  e.  { x  |  ph }
)
15 peano5 4695 . . . 4  |-  ( (
(/)  e.  { x  |  ph }  /\  A. y  e.  om  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )  ->  om  C_  { x  |  ph } )
165, 14, 15mp2an 653 . . 3  |-  om  C_  { x  |  ph }
1716sseli 3189 . 2  |-  ( A  e.  om  ->  A  e.  { x  |  ph } )
18 finds.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
1918elabg 2928 . 2  |-  ( A  e.  om  ->  ( A  e.  { x  |  ph }  <->  ta )
)
2017, 19mpbid 201 1  |-  ( A  e.  om  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556    C_ wss 3165   (/)c0 3468   suc csuc 4410   omcom 4672
This theorem is referenced by:  findsg  4699  findes  4702  seqomlem1  6478  nna0r  6623  nnm0r  6624  nnawordi  6635  nneob  6666  nneneq  7060  pssnn  7097  inf3lem1  7345  inf3lem2  7346  cantnfval2  7386  cantnfp1lem3  7398  r1fin  7461  ackbij1lem14  7875  ackbij1lem16  7877  ackbij1  7880  ackbij2lem2  7882  ackbij2lem3  7883  infpssrlem4  7948  fin23lem14  7975  fin23lem34  7988  itunitc1  8062  ituniiun  8064  om2uzuzi  11028  om2uzlti  11029  om2uzrdg  11035  uzrdgxfr  11045  hashgadd  11375  mreexexd  13566  trpredmintr  24305  findfvcl  24963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673
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