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Theorem tfinds 6593
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
tfinds.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
tfinds.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
tfinds.3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
tfinds.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
tfinds.5  |-  ps
tfinds.6  |-  ( y  e.  On  ->  ( ch  ->  th ) )
tfinds.7  |-  ( Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) )
Assertion
Ref Expression
tfinds  |-  ( A  e.  On  ->  ta )
Distinct variable groups:    x, y    x, A    ch, x    ta, x    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    ch( y)    th( x, y)    ta( y)    A( y)

Proof of Theorem tfinds
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tfinds.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
2 tfinds.4 . 2  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
3 dflim3 6581 . . . . 5  |-  ( Lim  x  <->  ( Ord  x  /\  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
43notbii 294 . . . 4  |-  ( -. 
Lim  x  <->  -.  ( Ord  x  /\  -.  (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
5 iman 422 . . . . 5  |-  ( ( Ord  x  ->  (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  <->  -.  ( Ord  x  /\  -.  ( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) ) )
6 eloni 4802 . . . . . . 7  |-  ( x  e.  On  ->  Ord  x )
7 pm2.27 39 . . . . . . 7  |-  ( Ord  x  ->  ( ( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  -> 
( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) ) )
86, 7syl 16 . . . . . 6  |-  ( x  e.  On  ->  (
( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
9 tfinds.5 . . . . . . . . 9  |-  ps
10 tfinds.1 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
119, 10mpbiri 233 . . . . . . . 8  |-  ( x  =  (/)  ->  ph )
1211a1d 25 . . . . . . 7  |-  ( x  =  (/)  ->  ( A. y  e.  x  ch  ->  ph ) )
13 nfra1 2763 . . . . . . . . 9  |-  F/ y A. y  e.  x  ch
14 nfv 1715 . . . . . . . . 9  |-  F/ y
ph
1513, 14nfim 1928 . . . . . . . 8  |-  F/ y ( A. y  e.  x  ch  ->  ph )
16 vex 3037 . . . . . . . . . . . . 13  |-  y  e. 
_V
1716sucid 4871 . . . . . . . . . . . 12  |-  y  e. 
suc  y
181rspcv 3131 . . . . . . . . . . . 12  |-  ( y  e.  suc  y  -> 
( A. x  e. 
suc  y ph  ->  ch ) )
1917, 18ax-mp 5 . . . . . . . . . . 11  |-  ( A. x  e.  suc  y ph  ->  ch )
20 tfinds.6 . . . . . . . . . . 11  |-  ( y  e.  On  ->  ( ch  ->  th ) )
2119, 20syl5 32 . . . . . . . . . 10  |-  ( y  e.  On  ->  ( A. x  e.  suc  y ph  ->  th )
)
22 raleq 2979 . . . . . . . . . . . 12  |-  ( x  =  suc  y  -> 
( A. z  e.  x  [ z  /  x ] ph  <->  A. z  e.  suc  y [ z  /  x ] ph ) )
23 nfv 1715 . . . . . . . . . . . . . . 15  |-  F/ x ch
2423, 1sbie 2153 . . . . . . . . . . . . . 14  |-  ( [ y  /  x ] ph 
<->  ch )
25 sbequ 2121 . . . . . . . . . . . . . 14  |-  ( y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
2624, 25syl5bbr 259 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  ( ch 
<->  [ z  /  x ] ph ) )
2726cbvralv 3009 . . . . . . . . . . . 12  |-  ( A. y  e.  x  ch  <->  A. z  e.  x  [
z  /  x ] ph )
28 cbvralsv 3020 . . . . . . . . . . . 12  |-  ( A. x  e.  suc  y ph  <->  A. z  e.  suc  y [ z  /  x ] ph )
2922, 27, 283bitr4g 288 . . . . . . . . . . 11  |-  ( x  =  suc  y  -> 
( A. y  e.  x  ch  <->  A. x  e.  suc  y ph )
)
3029imbi1d 315 . . . . . . . . . 10  |-  ( x  =  suc  y  -> 
( ( A. y  e.  x  ch  ->  th )  <->  ( A. x  e.  suc  y ph  ->  th ) ) )
3121, 30syl5ibrcom 222 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( A. y  e.  x  ch  ->  th )
) )
32 tfinds.3 . . . . . . . . . . 11  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
3332biimprd 223 . . . . . . . . . 10  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
3433a1i 11 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( th  ->  ph )
) )
3531, 34syldd 66 . . . . . . . 8  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( A. y  e.  x  ch  ->  ph )
) )
3615, 35rexlimi 2864 . . . . . . 7  |-  ( E. y  e.  On  x  =  suc  y  ->  ( A. y  e.  x  ch  ->  ph ) )
3712, 36jaoi 377 . . . . . 6  |-  ( ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y )  -> 
( A. y  e.  x  ch  ->  ph )
)
388, 37syl6 33 . . . . 5  |-  ( x  e.  On  ->  (
( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  ->  ( A. y  e.  x  ch  ->  ph ) ) )
395, 38syl5bir 218 . . . 4  |-  ( x  e.  On  ->  ( -.  ( Ord  x  /\  -.  ( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) )  ->  ( A. y  e.  x  ch  ->  ph ) ) )
404, 39syl5bi 217 . . 3  |-  ( x  e.  On  ->  ( -.  Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) ) )
41 tfinds.7 . . 3  |-  ( Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) )
4240, 41pm2.61d2 160 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  ch  ->  ph ) )
431, 2, 42tfis3 6591 1  |-  ( A  e.  On  ->  ta )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1399   [wsb 1747    e. wcel 1826   A.wral 2732   E.wrex 2733   (/)c0 3711   Ord word 4791   Oncon0 4792   Lim wlim 4793   suc csuc 4794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-tr 4461  df-eprel 4705  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798
This theorem is referenced by:  tfindsg  6594  tfindes  6596  tfinds3  6598  oa0r  7106  om0r  7107  om1r  7110  oe1m  7112  oeoalem  7163  r1sdom  8105  r1tr  8107  alephon  8363  alephcard  8364  alephordi  8368  rdgprc  29392
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