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Theorem tfis 4326
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 3080 . . . . 5  |-  { x  e.  On  |  ph }  C_  On
2 nfcv 2220 . . . . . . 7  |-  F/_ x
z
3 nfrab1 2534 . . . . . . . . 9  |-  F/_ x { x  e.  On  |  ph }
42, 3nfss 2993 . . . . . . . 8  |-  F/ x  z  C_  { x  e.  On  |  ph }
53nfcri 2214 . . . . . . . 8  |-  F/ x  z  e.  { x  e.  On  |  ph }
64, 5nfim 1505 . . . . . . 7  |-  F/ x
( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
)
7 dfss3 2990 . . . . . . . . 9  |-  ( x 
C_  { x  e.  On  |  ph }  <->  A. y  e.  x  y  e.  { x  e.  On  |  ph }
)
8 sseq1 3021 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  C_  { x  e.  On  |  ph }  <->  z 
C_  { x  e.  On  |  ph }
) )
97, 8syl5bbr 192 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  <->  z 
C_  { x  e.  On  |  ph }
) )
10 rabid 2530 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  ph }  <->  ( x  e.  On  /\  ph ) )
11 eleq1 2142 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  e.  { x  e.  On  |  ph }  <->  z  e.  { x  e.  On  |  ph }
) )
1210, 11syl5bbr 192 . . . . . . . 8  |-  ( x  =  z  ->  (
( x  e.  On  /\ 
ph )  <->  z  e.  { x  e.  On  |  ph } ) )
139, 12imbi12d 232 . . . . . . 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 1762 . . . . . . . . . . . 12  |-  ( w  =  y  ->  ( [ w  /  x ] ph  <->  [ y  /  x ] ph ) )
15 nfcv 2220 . . . . . . . . . . . . 13  |-  F/_ x On
16 nfcv 2220 . . . . . . . . . . . . 13  |-  F/_ w On
17 nfv 1462 . . . . . . . . . . . . 13  |-  F/ w ph
18 nfs1v 1857 . . . . . . . . . . . . 13  |-  F/ x [ w  /  x ] ph
19 sbequ12 1695 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
2015, 16, 17, 18, 19cbvrab 2600 . . . . . . . . . . . 12  |-  { x  e.  On  |  ph }  =  { w  e.  On  |  [ w  /  x ] ph }
2114, 20elrab2 2752 . . . . . . . . . . 11  |-  ( y  e.  { x  e.  On  |  ph }  <->  ( y  e.  On  /\  [ y  /  x ] ph ) )
2221simprbi 269 . . . . . . . . . 10  |-  ( y  e.  { x  e.  On  |  ph }  ->  [ y  /  x ] ph )
2322ralimi 2427 . . . . . . . . 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 32 . . . . . . . 8  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  ->  ph ) )
2625anc2li 322 . . . . . . 7  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  { x  e.  On  |  ph }  ->  ( x  e.  On  /\ 
ph ) ) )
272, 6, 13, 26vtoclgaf 2664 . . . . . 6  |-  ( z  e.  On  ->  (
z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph }
) )
2827rgen 2417 . . . . 5  |-  A. z  e.  On  ( z  C_  { x  e.  On  |  ph }  ->  z  e.  { x  e.  On  |  ph } )
29 tfi 4325 . . . . 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 417 . . . 4  |-  { x  e.  On  |  ph }  =  On
3130eqcomi 2086 . . 3  |-  On  =  { x  e.  On  |  ph }
3231rabeq2i 2599 . 2  |-  ( x  e.  On  <->  ( x  e.  On  /\  ph )
)
3332simprbi 269 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   [wsb 1686   A.wral 2349   {crab 2353    C_ wss 2974   Oncon0 4120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4282
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-in 2980  df-ss 2987  df-uni 3604  df-tr 3878  df-iord 4123  df-on 4125
This theorem is referenced by:  tfis2f  4327
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