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Theorem scottex 7557
Description: Scott's trick collects all sets that have a certain property and are of the 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 4152 . . . 4  |-  (/)  e.  _V
2 eleq1 2345 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 224 . . 3  |-  ( A  =  (/)  ->  A  e. 
_V )
4 rabexg 4166 . . 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 3467 . . 3  |-  ( -.  A  =  (/)  <->  E. y 
y  e.  A )
7 nfra1 2595 . . . . . 6  |-  F/ y A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)
8 nfcv 2421 . . . . . 6  |-  F/_ y A
97, 8nfrab 2723 . . . . 5  |-  F/_ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }
109nfel1 2431 . . . 4  |-  F/ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V
11 rsp 2605 . . . . . . . 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 2629 . . . . . 6  |-  ( y  e.  A  ->  A. x  e.  A  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
14 ss2rab 3251 . . . . . 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 7469 . . . . . . . 8  |-  ( rank `  y )  e.  On
17 fveq2 5527 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( rank `  x )  =  ( rank `  w
) )
1817sseq1d 3207 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  w
)  C_  ( rank `  y ) ) )
1918elrab 2925 . . . . . . . . . 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 2610 . . . . . . . 8  |-  A. w  e.  { x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  ( rank `  y )
22 sseq2 3202 . . . . . . . . . 10  |-  ( z  =  ( rank `  y
)  ->  ( ( rank `  w )  C_  z 
<->  ( rank `  w
)  C_  ( rank `  y ) ) )
2322ralbidv 2565 . . . . . . . . 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 2886 . . . . . . . 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 7515 . . . . . . 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 4160 . . . . 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 1803 . . 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 1530    = wceq 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   {crab 2549   _Vcvv 2790    C_ wss 3154   (/)c0 3457   Oncon0 4394   ` cfv 5257   rankcrnk 7437
This theorem is referenced by:  scottexs  7559  cplem2  7562  kardex  7566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-reg 7308  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-r1 7438  df-rank 7439
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