MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  scottex Structured version   Visualization version   Unicode version

Theorem scottex 8356
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 4535 . . . 4  |-  (/)  e.  _V
2 eleq1 2517 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 237 . . 3  |-  ( A  =  (/)  ->  A  e. 
_V )
4 rabexg 4553 . . 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 3742 . . 3  |-  ( -.  A  =  (/)  <->  E. y 
y  e.  A )
7 nfra1 2769 . . . . . 6  |-  F/ y A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)
8 nfcv 2592 . . . . . 6  |-  F/_ y A
97, 8nfrab 2972 . . . . 5  |-  F/_ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }
109nfel1 2606 . . . 4  |-  F/ y { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V
11 rsp 2754 . . . . . . . 8  |-  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( y  e.  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1211com12 32 . . . . . . 7  |-  ( y  e.  A  ->  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1312ralrimivw 2803 . . . . . 6  |-  ( y  e.  A  ->  A. x  e.  A  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
14 ss2rab 3505 . . . . . 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 216 . . . . 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 8266 . . . . . . . 8  |-  ( rank `  y )  e.  On
17 fveq2 5865 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( rank `  x )  =  ( rank `  w
) )
1817sseq1d 3459 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  w
)  C_  ( rank `  y ) ) )
1918elrab 3196 . . . . . . . . . 10  |-  ( w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  <-> 
( w  e.  A  /\  ( rank `  w
)  C_  ( rank `  y ) ) )
2019simprbi 466 . . . . . . . . 9  |-  ( w  e.  { x  e.  A  |  ( rank `  x )  C_  ( rank `  y ) }  ->  ( rank `  w
)  C_  ( rank `  y ) )
2120rgen 2747 . . . . . . . 8  |-  A. w  e.  { x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  ( rank `  y )
22 sseq2 3454 . . . . . . . . . 10  |-  ( z  =  ( rank `  y
)  ->  ( ( rank `  w )  C_  z 
<->  ( rank `  w
)  C_  ( rank `  y ) ) )
2322ralbidv 2827 . . . . . . . . 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 3150 . . . . . . . 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 678 . . . . . . 7  |-  E. z  e.  On  A. w  e. 
{ x  e.  A  |  ( rank `  x
)  C_  ( rank `  y ) }  ( rank `  w )  C_  z
26 bndrank 8312 . . . . . . 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 4547 . . . . 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 1995 . . 3  |-  ( E. y  y  e.  A  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  e.  _V )
316, 30sylbi 199 . 2  |-  ( -.  A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V )
325, 31pm2.61i 168 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 1444   E.wex 1663    e. wcel 1887   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045    C_ wss 3404   (/)c0 3731   Oncon0 5423   ` cfv 5582   rankcrnk 8234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-r1 8235  df-rank 8236
This theorem is referenced by:  scottexs  8358  cplem2  8361  kardex  8365  scottexf  32411
  Copyright terms: Public domain W3C validator