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Theorem scottex 8320
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 4587 . . . 4  |-  (/)  e.  _V
2 eleq1 2529 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 233 . . 3  |-  ( A  =  (/)  ->  A  e. 
_V )
4 rabexg 4606 . . 3  |-  ( A  e.  _V  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
53, 4syl 16 . 2  |-  ( A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
6 neq0 3804 . . 3  |-  ( -.  A  =  (/)  <->  E. y 
y  e.  A )
7 nfra1 2838 . . . . . 6  |-  F/ y A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)
8 nfcv 2619 . . . . . 6  |-  F/_ y A
97, 8nfrab 3039 . . . . 5  |-  F/_ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }
109nfel1 2635 . . . 4  |-  F/ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V
11 rsp 2823 . . . . . . . 8  |-  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( y  e.  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1211com12 31 . . . . . . 7  |-  ( y  e.  A  ->  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1312ralrimivw 2872 . . . . . 6  |-  ( y  e.  A  ->  A. x  e.  A  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
14 ss2rab 3572 . . . . . 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 212 . . . . 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 8230 . . . . . . . 8  |-  ( rank `  y )  e.  On
17 fveq2 5872 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( rank `  x )  =  ( rank `  w
) )
1817sseq1d 3526 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  w
)  C_  ( rank `  y ) ) )
1918elrab 3257 . . . . . . . . . 10  |-  ( w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  <-> 
( w  e.  A  /\  ( rank `  w
)  C_  ( rank `  y ) ) )
2019simprbi 464 . . . . . . . . 9  |-  ( w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ->  ( rank `  w
)  C_  ( rank `  y ) )
2120rgen 2817 . . . . . . . 8  |-  A. w  e.  { x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  ( rank `  y )
22 sseq2 3521 . . . . . . . . . 10  |-  ( z  =  ( rank `  y
)  ->  ( ( rank `  w )  C_  z 
<->  ( rank `  w
)  C_  ( rank `  y ) ) )
2322ralbidv 2896 . . . . . . . . 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 3210 . . . . . . . 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 672 . . . . . . 7  |-  E. z  e.  On  A. w  e. 
{ x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  z
26 bndrank 8276 . . . . . . 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 5 . . . . . 6  |-  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
2827ssex 4600 . . . . 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 16 . . . 4  |-  ( y  e.  A  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
3010, 29exlimi 1913 . . 3  |-  ( E. y  y  e.  A  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V )
316, 30sylbi 195 . 2  |-  ( -.  A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
325, 31pm2.61i 164 1  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1395   E.wex 1613    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   Oncon0 4887   ` cfv 5594   rankcrnk 8198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199  df-rank 8200
This theorem is referenced by:  scottexs  8322  cplem2  8325  kardex  8329  scottexf  30738
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