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Theorem tfis 4748
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 3344 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
2 nfcv 2502 . . . . . . 7  |-  F/_ x
z
3 nfrab1 2805 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  ph }
42, 3nfss 3259 . . . . . . . 8  |-  F/ x  z  C_  { x  e.  On  |  ph }
53nfcri 2496 . . . . . . . 8  |-  F/ x  z  e.  { x  e.  On  |  ph }
64, 5nfim 1820 . . . . . . 7  |-  F/ x
( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
)
7 dfss3 3256 . . . . . . . . 9  |-  ( x 
C_  { x  e.  On  |  ph }  <->  A. y  e.  x  y  e.  { x  e.  On  |  ph }
)
8 sseq1 3285 . . . . . . . . 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 2801 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  ph }  <->  ( x  e.  On  /\  ph ) )
11 eleq1 2426 . . . . . . . . 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 2073 . . . . . . . . . . . 12  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
15 nfcv 2502 . . . . . . . . . . . . 13  |-  F/_ x On
16 nfcv 2502 . . . . . . . . . . . . 13  |-  F/_ w On
17 nfv 1624 . . . . . . . . . . . . 13  |-  F/ w ph
18 nfs1v 2119 . . . . . . . . . . . . 13  |-  F/ x [ w  /  x ] ph
19 sbequ12 1931 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
2015, 16, 17, 18, 19cbvrab 2871 . . . . . . . . . . . 12  |-  { x  e.  On  |  ph }  =  { w  e.  On  |  [ w  /  x ] ph }
2114, 20elrab2 3011 . . . . . . . . . . 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 2703 . . . . . . . . 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 2933 . . . . . 6  |-  ( z  e.  On  ->  (
z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
) )
2827rgen 2693 . . . . 5  |-  A. z  e.  On  ( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph } )
29 tfi 4747 . . . . 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 2370 . . 3  |-  On  =  { x  e.  On  |  ph }
3231rabeq2i 2870 . 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 1647   [wsb 1653    e. wcel 1715   A.wral 2628   {crab 2632    C_ wss 3238   Oncon0 4495
This theorem is referenced by:  tfis2f  4749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499
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