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Theorem scott0s 7558
Description: Theorem scheme version of scott0 7556. 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 3473 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 scott0 7556 . . . 4  |-  ( { x  |  ph }  =  (/)  <->  { z  e.  {
x  |  ph }  |  A. y  e.  {
x  |  ph } 
( rank `  z )  C_  ( rank `  y
) }  =  (/) )
3 nfcv 2419 . . . . . . 7  |-  F/_ z { x  |  ph }
4 nfab1 2421 . . . . . . 7  |-  F/_ x { x  |  ph }
5 nfv 1605 . . . . . . . 8  |-  F/ x
( rank `  z )  C_  ( rank `  y
)
64, 5nfral 2596 . . . . . . 7  |-  F/ x A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y )
7 nfv 1605 . . . . . . 7  |-  F/ z A. y  e.  {
x  |  ph } 
( rank `  x )  C_  ( rank `  y
)
8 fveq2 5525 . . . . . . . . 9  |-  ( z  =  x  ->  ( rank `  z )  =  ( rank `  x
) )
98sseq1d 3205 . . . . . . . 8  |-  ( z  =  x  ->  (
( rank `  z )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
109ralbidv 2563 . . . . . . 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 2786 . . . . . 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 2552 . . . . . 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 2271 . . . . . . . 8  |-  ( x  e.  { x  | 
ph }  <->  ph )
14 df-ral 2548 . . . . . . . . 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 2992 . . . . . . . . . . 11  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
1615imbi1i 315 . . . . . . . . . 10  |-  ( (
[. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) )  <-> 
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1716albii 1553 . . . . . . . . 9  |-  ( A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) )  <->  A. y
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1814, 17bitr4i 243 . . . . . . . 8  |-  ( A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y )  <->  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1913, 18anbi12i 678 . . . . . . 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 2395 . . . . . 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 2307 . . . . 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 2290 . . . 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 240 . . 3  |-  ( { x  |  ph }  =  (/)  <->  { x  |  (
ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) ) }  =  (/) )
2423necon3bii 2478 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  { x  |  (
ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) ) }  =/=  (/) )
251, 24bitr3i 242 1  |-  ( E. x ph  <->  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   {crab 2547   [.wsbc 2991    C_ wss 3152   (/)c0 3455   ` cfv 5255   rankcrnk 7435
This theorem is referenced by:  hta  7567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
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