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Theorem finds 6507
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 step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (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 4427 . . . . . 6  |-  (/)  e.  _V
3 finds.1 . . . . . 6  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
42, 3elab 3111 . . . . 5  |-  ( (/)  e.  { x  |  ph } 
<->  ps )
51, 4mpbir 209 . . . 4  |-  (/)  e.  {
x  |  ph }
6 finds.6 . . . . . 6  |-  ( y  e.  om  ->  ( ch  ->  th ) )
7 vex 2980 . . . . . . 7  |-  y  e. 
_V
8 finds.2 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
97, 8elab 3111 . . . . . 6  |-  ( y  e.  { x  | 
ph }  <->  ch )
107sucex 6427 . . . . . . 7  |-  suc  y  e.  _V
11 finds.3 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
1210, 11elab 3111 . . . . . 6  |-  ( suc  y  e.  { x  |  ph }  <->  th )
136, 9, 123imtr4g 270 . . . . 5  |-  ( y  e.  om  ->  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )
1413rgen 2786 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ph }  ->  suc  y  e.  { x  |  ph }
)
15 peano5 6504 . . . 4  |-  ( (
(/)  e.  { x  |  ph }  /\  A. y  e.  om  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )  ->  om  C_  { x  |  ph } )
165, 14, 15mp2an 672 . . 3  |-  om  C_  { x  |  ph }
1716sseli 3357 . 2  |-  ( A  e.  om  ->  A  e.  { x  |  ph } )
18 finds.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
1918elabg 3112 . 2  |-  ( A  e.  om  ->  ( A  e.  { x  |  ph }  <->  ta )
)
2017, 19mpbid 210 1  |-  ( A  e.  om  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720    C_ wss 3333   (/)c0 3642   suc csuc 4726   omcom 6481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-om 6482
This theorem is referenced by:  findsg  6508  findes  6511  seqomlem1  6910  nna0r  7053  nnm0r  7054  nnawordi  7065  nneob  7096  nneneq  7499  pssnn  7536  inf3lem1  7839  inf3lem2  7840  cantnfval2  7882  cantnfp1lem3  7893  cantnfval2OLD  7912  cantnfp1lem3OLD  7919  r1fin  7985  ackbij1lem14  8407  ackbij1lem16  8409  ackbij1  8412  ackbij2lem2  8414  ackbij2lem3  8415  infpssrlem4  8480  fin23lem14  8507  fin23lem34  8520  itunitc1  8594  ituniiun  8596  om2uzuzi  11777  om2uzlti  11778  om2uzrdg  11784  uzrdgxfr  11794  hashgadd  12145  mreexexd  14591  trpredmintr  27700  findfvcl  28303
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