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Theorem tfis 4834
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 3428 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
2 nfcv 2572 . . . . . . 7  |-  F/_ x
z
3 nfrab1 2888 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  ph }
42, 3nfss 3341 . . . . . . . 8  |-  F/ x  z  C_  { x  e.  On  |  ph }
53nfcri 2566 . . . . . . . 8  |-  F/ x  z  e.  { x  e.  On  |  ph }
64, 5nfim 1832 . . . . . . 7  |-  F/ x
( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
)
7 dfss3 3338 . . . . . . . . 9  |-  ( x 
C_  { x  e.  On  |  ph }  <->  A. y  e.  x  y  e.  { x  e.  On  |  ph }
)
8 sseq1 3369 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  C_  { x  e.  On  |  ph }  <->  z 
C_  { x  e.  On  |  ph }
) )
97, 8syl5bbr 251 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  <->  z 
C_  { x  e.  On  |  ph }
) )
10 rabid 2884 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  ph }  <->  ( x  e.  On  /\  ph ) )
11 eleq1 2496 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  e.  { x  e.  On  |  ph }  <->  z  e.  { x  e.  On  |  ph }
) )
1210, 11syl5bbr 251 . . . . . . . 8  |-  ( x  =  z  ->  (
( x  e.  On  /\ 
ph )  <->  z  e.  { x  e.  On  |  ph } ) )
139, 12imbi12d 312 . . . . . . 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 2111 . . . . . . . . . . . 12  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
15 nfcv 2572 . . . . . . . . . . . . 13  |-  F/_ x On
16 nfcv 2572 . . . . . . . . . . . . 13  |-  F/_ w On
17 nfv 1629 . . . . . . . . . . . . 13  |-  F/ w ph
18 nfs1v 2182 . . . . . . . . . . . . 13  |-  F/ x [ w  /  x ] ph
19 sbequ12 1944 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
2015, 16, 17, 18, 19cbvrab 2954 . . . . . . . . . . . 12  |-  { x  e.  On  |  ph }  =  { w  e.  On  |  [ w  /  x ] ph }
2114, 20elrab2 3094 . . . . . . . . . . 11  |-  ( y  e.  { x  e.  On  |  ph }  <->  ( y  e.  On  /\  [ y  /  x ] ph ) )
2221simprbi 451 . . . . . . . . . 10  |-  ( y  e.  { x  e.  On  |  ph }  ->  [ y  /  x ] ph )
2322ralimi 2781 . . . . . . . . 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 30 . . . . . . . 8  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  ->  ph ) )
2625anc2li 541 . . . . . . 7  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  ->  ( x  e.  On  /\ 
ph ) ) )
272, 6, 13, 26vtoclgaf 3016 . . . . . 6  |-  ( z  e.  On  ->  (
z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
) )
2827rgen 2771 . . . . 5  |-  A. z  e.  On  ( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph } )
29 tfi 4833 . . . . 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 654 . . . 4  |-  { x  e.  On  |  ph }  =  On
3130eqcomi 2440 . . 3  |-  On  =  { x  e.  On  |  ph }
3231rabeq2i 2953 . 2  |-  ( x  e.  On  <->  ( x  e.  On  /\  ph )
)
3332simprbi 451 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   [wsb 1658    e. wcel 1725   A.wral 2705   {crab 2709    C_ wss 3320   Oncon0 4581
This theorem is referenced by:  tfis2f  4835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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