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Theorem scott0 8205
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, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e.  A is empty). (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
scott0  |-  ( A  =  (/)  <->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/) )
Distinct variable group:    x, y, A

Proof of Theorem scott0
StepHypRef Expression
1 rabeq 3072 . . 3  |-  ( A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  { x  e.  (/)  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) } )
2 rab0 3767 . . 3  |-  { x  e.  (/)  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/)
31, 2syl6eq 2511 . 2  |-  ( A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/) )
4 n0 3755 . . . . . . . 8  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 nfre1 2891 . . . . . . . . 9  |-  F/ x E. x  e.  A  ( rank `  x )  =  ( rank `  x
)
6 eqid 2454 . . . . . . . . . 10  |-  ( rank `  x )  =  (
rank `  x )
7 rspe 2895 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) )  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
86, 7mpan2 671 . . . . . . . . 9  |-  ( x  e.  A  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
95, 8exlimi 1850 . . . . . . . 8  |-  ( E. x  x  e.  A  ->  E. x  e.  A  ( rank `  x )  =  ( rank `  x
) )
104, 9sylbi 195 . . . . . . 7  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
11 fvex 5810 . . . . . . . . . . 11  |-  ( rank `  x )  e.  _V
12 eqeq1 2458 . . . . . . . . . . . 12  |-  ( y  =  ( rank `  x
)  ->  ( y  =  ( rank `  x
)  <->  ( rank `  x
)  =  ( rank `  x ) ) )
1312anbi2d 703 . . . . . . . . . . 11  |-  ( y  =  ( rank `  x
)  ->  ( (
x  e.  A  /\  y  =  ( rank `  x ) )  <->  ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) ) ) )
1411, 13spcev 3170 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) )  ->  E. y
( x  e.  A  /\  y  =  ( rank `  x ) ) )
1514eximi 1626 . . . . . . . . 9  |-  ( E. x ( x  e.  A  /\  ( rank `  x )  =  (
rank `  x )
)  ->  E. x E. y ( x  e.  A  /\  y  =  ( rank `  x
) ) )
16 excom 1789 . . . . . . . . 9  |-  ( E. y E. x ( x  e.  A  /\  y  =  ( rank `  x ) )  <->  E. x E. y ( x  e.  A  /\  y  =  ( rank `  x
) ) )
1715, 16sylibr 212 . . . . . . . 8  |-  ( E. x ( x  e.  A  /\  ( rank `  x )  =  (
rank `  x )
)  ->  E. y E. x ( x  e.  A  /\  y  =  ( rank `  x
) ) )
18 df-rex 2805 . . . . . . . 8  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  x
)  <->  E. x ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) ) )
19 df-rex 2805 . . . . . . . . 9  |-  ( E. x  e.  A  y  =  ( rank `  x
)  <->  E. x ( x  e.  A  /\  y  =  ( rank `  x
) ) )
2019exbii 1635 . . . . . . . 8  |-  ( E. y E. x  e.  A  y  =  (
rank `  x )  <->  E. y E. x ( x  e.  A  /\  y  =  ( rank `  x ) ) )
2117, 18, 203imtr4i 266 . . . . . . 7  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  x
)  ->  E. y E. x  e.  A  y  =  ( rank `  x ) )
2210, 21syl 16 . . . . . 6  |-  ( A  =/=  (/)  ->  E. y E. x  e.  A  y  =  ( rank `  x ) )
23 abn0 3765 . . . . . 6  |-  ( { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/)  <->  E. y E. x  e.  A  y  =  ( rank `  x )
)
2422, 23sylibr 212 . . . . 5  |-  ( A  =/=  (/)  ->  { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/) )
2511dfiin2 4314 . . . . . 6  |-  |^|_ x  e.  A  ( rank `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( rank `  x ) }
26 rankon 8114 . . . . . . . . . 10  |-  ( rank `  x )  e.  On
27 eleq1 2526 . . . . . . . . . 10  |-  ( y  =  ( rank `  x
)  ->  ( y  e.  On  <->  ( rank `  x
)  e.  On ) )
2826, 27mpbiri 233 . . . . . . . . 9  |-  ( y  =  ( rank `  x
)  ->  y  e.  On )
2928rexlimivw 2943 . . . . . . . 8  |-  ( E. x  e.  A  y  =  ( rank `  x
)  ->  y  e.  On )
3029abssi 3536 . . . . . . 7  |-  { y  |  E. x  e.  A  y  =  (
rank `  x ) }  C_  On
31 onint 6517 . . . . . . 7  |-  ( ( { y  |  E. x  e.  A  y  =  ( rank `  x
) }  C_  On  /\ 
{ y  |  E. x  e.  A  y  =  ( rank `  x
) }  =/=  (/) )  ->  |^| { y  |  E. x  e.  A  y  =  ( rank `  x
) }  e.  {
y  |  E. x  e.  A  y  =  ( rank `  x ) } )
3230, 31mpan 670 . . . . . 6  |-  ( { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/)  ->  |^| { y  |  E. x  e.  A  y  =  (
rank `  x ) }  e.  { y  |  E. x  e.  A  y  =  ( rank `  x ) } )
3325, 32syl5eqel 2546 . . . . 5  |-  ( { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/)  ->  |^|_ x  e.  A  ( rank `  x
)  e.  { y  |  E. x  e.  A  y  =  (
rank `  x ) } )
34 nfii1 4310 . . . . . . . . 9  |-  F/_ x |^|_ x  e.  A  (
rank `  x )
3534nfeq2 2633 . . . . . . . 8  |-  F/ x  y  =  |^|_ x  e.  A  ( rank `  x
)
36 eqeq1 2458 . . . . . . . 8  |-  ( y  =  |^|_ x  e.  A  ( rank `  x )  ->  ( y  =  (
rank `  x )  <->  |^|_
x  e.  A  (
rank `  x )  =  ( rank `  x
) ) )
3735, 36rexbid 2867 . . . . . . 7  |-  ( y  =  |^|_ x  e.  A  ( rank `  x )  ->  ( E. x  e.  A  y  =  (
rank `  x )  <->  E. x  e.  A  |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )
) )
3837elabg 3214 . . . . . 6  |-  ( |^|_ x  e.  A  ( rank `  x )  e.  {
y  |  E. x  e.  A  y  =  ( rank `  x ) }  ->  ( |^|_ x  e.  A  ( rank `  x )  e.  {
y  |  E. x  e.  A  y  =  ( rank `  x ) } 
<->  E. x  e.  A  |^|_
x  e.  A  (
rank `  x )  =  ( rank `  x
) ) )
3938ibi 241 . . . . 5  |-  ( |^|_ x  e.  A  ( rank `  x )  e.  {
y  |  E. x  e.  A  y  =  ( rank `  x ) }  ->  E. x  e.  A  |^|_
x  e.  A  (
rank `  x )  =  ( rank `  x
) )
40 ssid 3484 . . . . . . . . . 10  |-  ( rank `  y )  C_  ( rank `  y )
41 fveq2 5800 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( rank `  x )  =  ( rank `  y
) )
4241sseq1d 3492 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  y
)  C_  ( rank `  y ) ) )
4342rspcev 3179 . . . . . . . . . 10  |-  ( ( y  e.  A  /\  ( rank `  y )  C_  ( rank `  y
) )  ->  E. x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
4440, 43mpan2 671 . . . . . . . . 9  |-  ( y  e.  A  ->  E. x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
45 iinss 4330 . . . . . . . . 9  |-  ( E. x  e.  A  (
rank `  x )  C_  ( rank `  y
)  ->  |^|_ x  e.  A  ( rank `  x
)  C_  ( rank `  y ) )
4644, 45syl 16 . . . . . . . 8  |-  ( y  e.  A  ->  |^|_ x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
47 sseq1 3486 . . . . . . . 8  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  ( |^|_ x  e.  A  ( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
4846, 47syl5ib 219 . . . . . . 7  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  ( y  e.  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
4948ralrimiv 2828 . . . . . 6  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) )
5049reximi 2929 . . . . 5  |-  ( E. x  e.  A  |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  E. x  e.  A  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) )
5124, 33, 39, 504syl 21 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) )
52 rabn0 3766 . . . 4  |-  ( { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =/=  (/)  <->  E. x  e.  A  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) )
5351, 52sylibr 212 . . 3  |-  ( A  =/=  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =/=  (/) )
5453necon4i 2696 . 2  |-  ( { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/)  ->  A  =  (/) )
553, 54impbii 188 1  |-  ( A  =  (/)  <->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803    C_ wss 3437   (/)c0 3746   |^|cint 4237   |^|_ciin 4281   Oncon0 4828   ` cfv 5527   rankcrnk 8082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-recs 6943  df-rdg 6977  df-r1 8083  df-rank 8084
This theorem is referenced by:  scott0s  8207  cplem1  8208  karden  8214  scott0f  29130
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