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Theorem scottex 7555
Description: Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.)
Assertion
Ref Expression
scottex  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
Distinct variable group:    x, y, A

Proof of Theorem scottex
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4150 . . . 4  |-  (/)  e.  _V
2 eleq1 2343 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 224 . . 3  |-  ( A  =  (/)  ->  A  e. 
_V )
4 rabexg 4164 . . 3  |-  ( A  e.  _V  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
53, 4syl 15 . 2  |-  ( A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
6 neq0 3465 . . 3  |-  ( -.  A  =  (/)  <->  E. y 
y  e.  A )
7 nfra1 2593 . . . . . 6  |-  F/ y A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)
8 nfcv 2419 . . . . . 6  |-  F/_ y A
97, 8nfrab 2721 . . . . 5  |-  F/_ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }
109nfel1 2429 . . . 4  |-  F/ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V
11 rsp 2603 . . . . . . . 8  |-  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( y  e.  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1211com12 27 . . . . . . 7  |-  ( y  e.  A  ->  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1312ralrimivw 2627 . . . . . 6  |-  ( y  e.  A  ->  A. x  e.  A  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
14 ss2rab 3249 . . . . . 6  |-  ( { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  C_  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) } 
<-> 
A. x  e.  A  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1513, 14sylibr 203 . . . . 5  |-  ( y  e.  A  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  C_  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) } )
16 rankon 7467 . . . . . . . 8  |-  ( rank `  y )  e.  On
17 fveq2 5525 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( rank `  x )  =  ( rank `  w
) )
1817sseq1d 3205 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  w
)  C_  ( rank `  y ) ) )
1918elrab 2923 . . . . . . . . . 10  |-  ( w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  <-> 
( w  e.  A  /\  ( rank `  w
)  C_  ( rank `  y ) ) )
2019simprbi 450 . . . . . . . . 9  |-  ( w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ->  ( rank `  w
)  C_  ( rank `  y ) )
2120rgen 2608 . . . . . . . 8  |-  A. w  e.  { x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  ( rank `  y )
22 sseq2 3200 . . . . . . . . . 10  |-  ( z  =  ( rank `  y
)  ->  ( ( rank `  w )  C_  z 
<->  ( rank `  w
)  C_  ( rank `  y ) ) )
2322ralbidv 2563 . . . . . . . . 9  |-  ( z  =  ( rank `  y
)  ->  ( A. w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ( rank `  w
)  C_  z  <->  A. w  e.  { x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  ( rank `  y )
) )
2423rspcev 2884 . . . . . . . 8  |-  ( ( ( rank `  y
)  e.  On  /\  A. w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ( rank `  w
)  C_  ( rank `  y ) )  ->  E. z  e.  On  A. w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ( rank `  w
)  C_  z )
2516, 21, 24mp2an 653 . . . . . . 7  |-  E. z  e.  On  A. w  e. 
{ x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  z
26 bndrank 7513 . . . . . . 7  |-  ( E. z  e.  On  A. w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ( rank `  w
)  C_  z  ->  { x  e.  A  | 
( rank `  x )  C_  ( rank `  y
) }  e.  _V )
2725, 26ax-mp 8 . . . . . 6  |-  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
2827ssex 4158 . . . . 5  |-  ( { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  C_  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
2915, 28syl 15 . . . 4  |-  ( y  e.  A  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
3010, 29exlimi 1801 . . 3  |-  ( E. y  y  e.  A  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V )
316, 30sylbi 187 . 2  |-  ( -.  A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
325, 31pm2.61i 156 1  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   Oncon0 4392   ` cfv 5255   rankcrnk 7435
This theorem is referenced by:  scottexs  7557  cplem2  7560  kardex  7564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
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