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Theorem scott0s 8091
Description: Theorem scheme version of scott0 8089. 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 3653 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 scott0 8089 . . . 4  |-  ( { x  |  ph }  =  (/)  <->  { z  e.  {
x  |  ph }  |  A. y  e.  {
x  |  ph } 
( rank `  z )  C_  ( rank `  y
) }  =  (/) )
3 nfcv 2577 . . . . . . 7  |-  F/_ z { x  |  ph }
4 nfab1 2579 . . . . . . 7  |-  F/_ x { x  |  ph }
5 nfv 1678 . . . . . . . 8  |-  F/ x
( rank `  z )  C_  ( rank `  y
)
64, 5nfral 2767 . . . . . . 7  |-  F/ x A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y )
7 nfv 1678 . . . . . . 7  |-  F/ z A. y  e.  {
x  |  ph } 
( rank `  x )  C_  ( rank `  y
)
8 fveq2 5688 . . . . . . . . 9  |-  ( z  =  x  ->  ( rank `  z )  =  ( rank `  x
) )
98sseq1d 3380 . . . . . . . 8  |-  ( z  =  x  ->  (
( rank `  z )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
109ralbidv 2733 . . . . . . 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 2968 . . . . . 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 2722 . . . . . 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 2429 . . . . . . . 8  |-  ( x  e.  { x  | 
ph }  <->  ph )
14 df-ral 2718 . . . . . . . . 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 3184 . . . . . . . . . . 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 1615 . . . . . . . . 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 692 . . . . . . 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 2553 . . . . . 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 2465 . . . . 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 2638 . 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 1362    = wceq 1364   E.wex 1591    e. wcel 1761   {cab 2427    =/= wne 2604   A.wral 2713   {crab 2717   [.wsbc 3183    C_ wss 3325   (/)c0 3634   ` cfv 5415   rankcrnk 7966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862  df-r1 7967  df-rank 7968
This theorem is referenced by:  hta  8100
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