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Theorem scottex 7488
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
StepHypRef Expression
1 0ex 4090 . . . 4  |-  (/)  e.  _V
2 eleq1 2316 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 226 . . 3  |-  ( A  =  (/)  ->  A  e. 
_V )
4 rabexg 4104 . . 3  |-  ( A  e.  _V  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
53, 4syl 17 . 2  |-  ( A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
6 neq0 3407 . . 3  |-  ( -.  A  =  (/)  <->  E. y 
y  e.  A )
7 nfra1 2564 . . . . . 6  |-  F/ y A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)
8 nfcv 2392 . . . . . 6  |-  F/_ y A
97, 8nfrab 2689 . . . . 5  |-  F/_ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }
109nfel1 2402 . . . 4  |-  F/ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V
11 ra4 2574 . . . . . . . 8  |-  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( y  e.  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1211com12 29 . . . . . . 7  |-  ( y  e.  A  ->  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1312ralrimivw 2598 . . . . . 6  |-  ( y  e.  A  ->  A. x  e.  A  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
14 ss2rab 3191 . . . . . 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 205 . . . . 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 7400 . . . . . . . 8  |-  ( rank `  y )  e.  On
17 fveq2 5423 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( rank `  x )  =  ( rank `  w
) )
1817sseq1d 3147 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  w
)  C_  ( rank `  y ) ) )
1918elrab 2874 . . . . . . . . . 10  |-  ( w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  <-> 
( w  e.  A  /\  ( rank `  w
)  C_  ( rank `  y ) ) )
2019simprbi 452 . . . . . . . . 9  |-  ( w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ->  ( rank `  w
)  C_  ( rank `  y ) )
2120rgen 2579 . . . . . . . 8  |-  A. w  e.  { x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  ( rank `  y )
22 sseq2 3142 . . . . . . . . . 10  |-  ( z  =  ( rank `  y
)  ->  ( ( rank `  w )  C_  z 
<->  ( rank `  w
)  C_  ( rank `  y ) ) )
2322ralbidv 2534 . . . . . . . . 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 )
) )
2423rcla4ev 2835 . . . . . . . 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 656 . . . . . . 7  |-  E. z  e.  On  A. w  e. 
{ x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  z
26 bndrank 7446 . . . . . . 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 10 . . . . . 6  |-  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
2827ssex 4098 . . . . 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 17 . . . 4  |-  ( y  e.  A  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
3010, 29exlimi 1781 . . 3  |-  ( E. y  y  e.  A  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V )
316, 30sylbi 189 . 2  |-  ( -.  A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
325, 31pm2.61i 158 1  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   {crab 2519   _Vcvv 2740    C_ wss 3094   (/)c0 3397   Oncon0 4329   ` cfv 4638   rankcrnk 7368
This theorem is referenced by:  scottexs  7490  cplem2  7493  kardex  7497
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-reg 7239  ax-inf2 7275
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-recs 6321  df-rdg 6356  df-r1 7369  df-rank 7370
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