MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfis Unicode version

Theorem tfis 4661
Description: Transfinite Induction Schema. If all ordinal numbers less than a given number  x have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
Hypothesis
Ref Expression
tfis.1  |-  ( x  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph ) )
Assertion
Ref Expression
tfis  |-  ( x  e.  On  ->  ph )
Distinct variable groups:    ph, y    x, y
Allowed substitution hint:    ph( x)

Proof of Theorem tfis
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3271 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
2 nfcv 2432 . . . . . . 7  |-  F/_ x
z
3 nfrab1 2733 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  ph }
42, 3nfss 3186 . . . . . . . 8  |-  F/ x  z  C_  { x  e.  On  |  ph }
53nfcri 2426 . . . . . . . 8  |-  F/ x  z  e.  { x  e.  On  |  ph }
64, 5nfim 1781 . . . . . . 7  |-  F/ x
( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
)
7 dfss3 3183 . . . . . . . . 9  |-  ( x 
C_  { x  e.  On  |  ph }  <->  A. y  e.  x  y  e.  { x  e.  On  |  ph }
)
8 sseq1 3212 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  C_  { x  e.  On  |  ph }  <->  z 
C_  { x  e.  On  |  ph }
) )
97, 8syl5bbr 250 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  <->  z 
C_  { x  e.  On  |  ph }
) )
10 rabid 2729 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  ph }  <->  ( x  e.  On  /\  ph ) )
11 eleq1 2356 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  e.  { x  e.  On  |  ph }  <->  z  e.  { x  e.  On  |  ph }
) )
1210, 11syl5bbr 250 . . . . . . . 8  |-  ( x  =  z  ->  (
( x  e.  On  /\ 
ph )  <->  z  e.  { x  e.  On  |  ph } ) )
139, 12imbi12d 311 . . . . . . 7  |-  ( x  =  z  ->  (
( A. y  e.  x  y  e.  {
x  e.  On  |  ph }  ->  ( x  e.  On  /\  ph )
)  <->  ( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph } ) ) )
14 sbequ 2013 . . . . . . . . . . . 12  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
15 nfcv 2432 . . . . . . . . . . . . 13  |-  F/_ x On
16 nfcv 2432 . . . . . . . . . . . . 13  |-  F/_ w On
17 nfv 1609 . . . . . . . . . . . . 13  |-  F/ w ph
18 nfs1v 2058 . . . . . . . . . . . . 13  |-  F/ x [ w  /  x ] ph
19 sbequ12 1872 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
2015, 16, 17, 18, 19cbvrab 2799 . . . . . . . . . . . 12  |-  { x  e.  On  |  ph }  =  { w  e.  On  |  [ w  /  x ] ph }
2114, 20elrab2 2938 . . . . . . . . . . 11  |-  ( y  e.  { x  e.  On  |  ph }  <->  ( y  e.  On  /\  [ y  /  x ] ph ) )
2221simprbi 450 . . . . . . . . . 10  |-  ( y  e.  { x  e.  On  |  ph }  ->  [ y  /  x ] ph )
2322ralimi 2631 . . . . . . . . 9  |-  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  ->  A. y  e.  x  [ y  /  x ] ph )
24 tfis.1 . . . . . . . . 9  |-  ( x  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph ) )
2523, 24syl5 28 . . . . . . . 8  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  ->  ph ) )
2625anc2li 540 . . . . . . 7  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  ->  ( x  e.  On  /\ 
ph ) ) )
272, 6, 13, 26vtoclgaf 2861 . . . . . 6  |-  ( z  e.  On  ->  (
z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
) )
2827rgen 2621 . . . . 5  |-  A. z  e.  On  ( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph } )
29 tfi 4660 . . . . 5  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
A. z  e.  On  ( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
) )  ->  { x  e.  On  |  ph }  =  On )
301, 28, 29mp2an 653 . . . 4  |-  { x  e.  On  |  ph }  =  On
3130eqcomi 2300 . . 3  |-  On  =  { x  e.  On  |  ph }
3231rabeq2i 2798 . 2  |-  ( x  e.  On  <->  ( x  e.  On  /\  ph )
)
3332simprbi 450 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   [wsb 1638    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   Oncon0 4408
This theorem is referenced by:  tfis2f  4662
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-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
  Copyright terms: Public domain W3C validator