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Theorem tfis 4536
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
StepHypRef Expression
1 ssrab2 3179 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
2 nfcv 2385 . . . . . . 7  |-  F/_ x
z
3 nfrab1 2679 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  ph }
42, 3nfss 3096 . . . . . . . 8  |-  F/ x  z  C_  { x  e.  On  |  ph }
53nfcri 2379 . . . . . . . 8  |-  F/ x  z  e.  { x  e.  On  |  ph }
64, 5nfim 1735 . . . . . . 7  |-  F/ x
( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
)
7 dfss3 3093 . . . . . . . . 9  |-  ( x 
C_  { x  e.  On  |  ph }  <->  A. y  e.  x  y  e.  { x  e.  On  |  ph }
)
8 sseq1 3120 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  C_  { x  e.  On  |  ph }  <->  z 
C_  { x  e.  On  |  ph }
) )
97, 8syl5bbr 252 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  <->  z 
C_  { x  e.  On  |  ph }
) )
10 rabid 2675 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  ph }  <->  ( x  e.  On  /\  ph ) )
11 eleq1 2313 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  e.  { x  e.  On  |  ph }  <->  z  e.  { x  e.  On  |  ph }
) )
1210, 11syl5bbr 252 . . . . . . . 8  |-  ( x  =  z  ->  (
( x  e.  On  /\ 
ph )  <->  z  e.  { x  e.  On  |  ph } ) )
139, 12imbi12d 313 . . . . . . 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 1952 . . . . . . . . . . . 12  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
15 nfcv 2385 . . . . . . . . . . . . 13  |-  F/_ x On
16 nfcv 2385 . . . . . . . . . . . . 13  |-  F/_ w On
17 nfv 1629 . . . . . . . . . . . . 13  |-  F/ w ph
18 nfs1v 2066 . . . . . . . . . . . . 13  |-  F/ x [ w  /  x ] ph
19 sbequ12 1892 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
2015, 16, 17, 18, 19cbvrab 2725 . . . . . . . . . . . 12  |-  { x  e.  On  |  ph }  =  { w  e.  On  |  [ w  /  x ] ph }
2114, 20elrab2 2862 . . . . . . . . . . 11  |-  ( y  e.  { x  e.  On  |  ph }  <->  ( y  e.  On  /\  [ y  /  x ] ph ) )
2221simprbi 452 . . . . . . . . . 10  |-  ( y  e.  { x  e.  On  |  ph }  ->  [ y  /  x ] ph )
2322ralimi 2580 . . . . . . . . 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 542 . . . . . . 7  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  ->  ( x  e.  On  /\ 
ph ) ) )
272, 6, 13, 26vtoclgaf 2786 . . . . . 6  |-  ( z  e.  On  ->  (
z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
) )
2827rgen 2570 . . . . 5  |-  A. z  e.  On  ( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph } )
29 tfi 4535 . . . . 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 656 . . . 4  |-  { x  e.  On  |  ph }  =  On
3130eqcomi 2257 . . 3  |-  On  =  { x  e.  On  |  ph }
3231rabeq2i 2724 . 2  |-  ( x  e.  On  <->  ( x  e.  On  /\  ph )
)
3332simprbi 452 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   [wsb 1882   A.wral 2509   {crab 2512    C_ wss 3078   Oncon0 4285
This theorem is referenced by:  tfis2f  4537
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289
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