MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  scott0s Structured version   Unicode version

Theorem scott0s 8306
Description: Theorem scheme version of scott0 8304. The collection of all  x of minimum rank such that 
ph ( x ) is true, is not empty iff there is an  x such that  ph ( x ) holds. (Contributed by NM, 13-Oct-2003.)
Assertion
Ref Expression
scott0s  |-  ( E. x ph  <->  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }  =/=  (/) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem scott0s
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 abn0 3787 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 scott0 8304 . . . 4  |-  ( { x  |  ph }  =  (/)  <->  { z  e.  {
x  |  ph }  |  A. y  e.  {
x  |  ph } 
( rank `  z )  C_  ( rank `  y
) }  =  (/) )
3 nfcv 2603 . . . . . . 7  |-  F/_ z { x  |  ph }
4 nfab1 2605 . . . . . . 7  |-  F/_ x { x  |  ph }
5 nfv 1692 . . . . . . . 8  |-  F/ x
( rank `  z )  C_  ( rank `  y
)
64, 5nfral 2827 . . . . . . 7  |-  F/ x A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y )
7 nfv 1692 . . . . . . 7  |-  F/ z A. y  e.  {
x  |  ph } 
( rank `  x )  C_  ( rank `  y
)
8 fveq2 5853 . . . . . . . . 9  |-  ( z  =  x  ->  ( rank `  z )  =  ( rank `  x
) )
98sseq1d 3514 . . . . . . . 8  |-  ( z  =  x  ->  (
( rank `  z )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
109ralbidv 2880 . . . . . . 7  |-  ( z  =  x  ->  ( A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y )  <->  A. y  e.  { x  |  ph }  ( rank `  x
)  C_  ( rank `  y ) ) )
113, 4, 6, 7, 10cbvrab 3091 . . . . . 6  |-  { z  e.  { x  | 
ph }  |  A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y ) }  =  { x  e. 
{ x  |  ph }  |  A. y  e.  { x  |  ph }  ( rank `  x
)  C_  ( rank `  y ) }
12 df-rab 2800 . . . . . 6  |-  { x  e.  { x  |  ph }  |  A. y  e.  { x  |  ph }  ( rank `  x
)  C_  ( rank `  y ) }  =  { x  |  (
x  e.  { x  |  ph }  /\  A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y ) ) }
13 abid 2428 . . . . . . . 8  |-  ( x  e.  { x  | 
ph }  <->  ph )
14 df-ral 2796 . . . . . . . . 9  |-  ( A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y )  <->  A. y
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
15 df-sbc 3312 . . . . . . . . . . 11  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
1615imbi1i 325 . . . . . . . . . 10  |-  ( (
[. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) )  <-> 
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1716albii 1625 . . . . . . . . 9  |-  ( A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) )  <->  A. y
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1814, 17bitr4i 252 . . . . . . . 8  |-  ( A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y )  <->  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1913, 18anbi12i 697 . . . . . . 7  |-  ( ( x  e.  { x  |  ph }  /\  A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y ) )  <-> 
( ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) ) )
2019abbii 2575 . . . . . 6  |-  { x  |  ( x  e. 
{ x  |  ph }  /\  A. y  e. 
{ x  |  ph }  ( rank `  x
)  C_  ( rank `  y ) ) }  =  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }
2111, 12, 203eqtri 2474 . . . . 5  |-  { z  e.  { x  | 
ph }  |  A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y ) }  =  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }
2221eqeq1i 2448 . . . 4  |-  ( { z  e.  { x  |  ph }  |  A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y ) }  =  (/)  <->  { x  |  (
ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) ) }  =  (/) )
232, 22bitri 249 . . 3  |-  ( { x  |  ph }  =  (/)  <->  { x  |  (
ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) ) }  =  (/) )
2423necon3bii 2709 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  { x  |  (
ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) ) }  =/=  (/) )
251, 24bitr3i 251 1  |-  ( E. x ph  <->  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1379    = wceq 1381   E.wex 1597    e. wcel 1802   {cab 2426    =/= wne 2636   A.wral 2791   {crab 2795   [.wsbc 3311    C_ wss 3459   (/)c0 3768   ` cfv 5575   rankcrnk 8181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-iin 4315  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-om 6683  df-recs 7041  df-rdg 7075  df-r1 8182  df-rank 8183
This theorem is referenced by:  hta  8315
  Copyright terms: Public domain W3C validator