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Theorem scott0s 8116
Description: Theorem scheme version of scott0 8114. 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 3677 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 scott0 8114 . . . 4  |-  ( { x  |  ph }  =  (/)  <->  { z  e.  {
x  |  ph }  |  A. y  e.  {
x  |  ph } 
( rank `  z )  C_  ( rank `  y
) }  =  (/) )
3 nfcv 2589 . . . . . . 7  |-  F/_ z { x  |  ph }
4 nfab1 2591 . . . . . . 7  |-  F/_ x { x  |  ph }
5 nfv 1673 . . . . . . . 8  |-  F/ x
( rank `  z )  C_  ( rank `  y
)
64, 5nfral 2790 . . . . . . 7  |-  F/ x A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y )
7 nfv 1673 . . . . . . 7  |-  F/ z A. y  e.  {
x  |  ph } 
( rank `  x )  C_  ( rank `  y
)
8 fveq2 5712 . . . . . . . . 9  |-  ( z  =  x  ->  ( rank `  z )  =  ( rank `  x
) )
98sseq1d 3404 . . . . . . . 8  |-  ( z  =  x  ->  (
( rank `  z )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
109ralbidv 2756 . . . . . . 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 2991 . . . . . 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 2745 . . . . . 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 2431 . . . . . . . 8  |-  ( x  e.  { x  | 
ph }  <->  ph )
14 df-ral 2741 . . . . . . . . 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 3208 . . . . . . . . . . 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 1610 . . . . . . . . 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 2561 . . . . . 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 2467 . . . . 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 2450 . . . 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 2634 . 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 1367    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2620   A.wral 2736   {crab 2740   [.wsbc 3207    C_ wss 3349   (/)c0 3658   ` cfv 5439   rankcrnk 7991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-om 6498  df-recs 6853  df-rdg 6887  df-r1 7992  df-rank 7993
This theorem is referenced by:  hta  8125
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