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Theorem scott0 7766
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 2910 . . 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 3608 . . 3  |-  { x  e.  (/)  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/)
31, 2syl6eq 2452 . 2  |-  ( A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/) )
4 n0 3597 . . . . . . . . 9  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 nfre1 2722 . . . . . . . . . 10  |-  F/ x E. x  e.  A  ( rank `  x )  =  ( rank `  x
)
6 eqid 2404 . . . . . . . . . . 11  |-  ( rank `  x )  =  (
rank `  x )
7 rspe 2727 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) )  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
86, 7mpan2 653 . . . . . . . . . 10  |-  ( x  e.  A  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
95, 8exlimi 1817 . . . . . . . . 9  |-  ( E. x  x  e.  A  ->  E. x  e.  A  ( rank `  x )  =  ( rank `  x
) )
104, 9sylbi 188 . . . . . . . 8  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
11 fvex 5701 . . . . . . . . . . . 12  |-  ( rank `  x )  e.  _V
12 eqeq1 2410 . . . . . . . . . . . . 13  |-  ( y  =  ( rank `  x
)  ->  ( y  =  ( rank `  x
)  <->  ( rank `  x
)  =  ( rank `  x ) ) )
1312anbi2d 685 . . . . . . . . . . . 12  |-  ( y  =  ( rank `  x
)  ->  ( (
x  e.  A  /\  y  =  ( rank `  x ) )  <->  ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) ) ) )
1411, 13spcev 3003 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) )  ->  E. y
( x  e.  A  /\  y  =  ( rank `  x ) ) )
1514eximi 1582 . . . . . . . . . 10  |-  ( E. x ( x  e.  A  /\  ( rank `  x )  =  (
rank `  x )
)  ->  E. x E. y ( x  e.  A  /\  y  =  ( rank `  x
) ) )
16 excom 1752 . . . . . . . . . 10  |-  ( E. y E. x ( x  e.  A  /\  y  =  ( rank `  x ) )  <->  E. x E. y ( x  e.  A  /\  y  =  ( rank `  x
) ) )
1715, 16sylibr 204 . . . . . . . . 9  |-  ( E. x ( x  e.  A  /\  ( rank `  x )  =  (
rank `  x )
)  ->  E. y E. x ( x  e.  A  /\  y  =  ( rank `  x
) ) )
18 df-rex 2672 . . . . . . . . 9  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  x
)  <->  E. x ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) ) )
19 df-rex 2672 . . . . . . . . . 10  |-  ( E. x  e.  A  y  =  ( rank `  x
)  <->  E. x ( x  e.  A  /\  y  =  ( rank `  x
) ) )
2019exbii 1589 . . . . . . . . 9  |-  ( E. y E. x  e.  A  y  =  (
rank `  x )  <->  E. y E. x ( x  e.  A  /\  y  =  ( rank `  x ) ) )
2117, 18, 203imtr4i 258 . . . . . . . 8  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  x
)  ->  E. y E. x  e.  A  y  =  ( rank `  x ) )
2210, 21syl 16 . . . . . . 7  |-  ( A  =/=  (/)  ->  E. y E. x  e.  A  y  =  ( rank `  x ) )
23 abn0 3606 . . . . . . 7  |-  ( { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/)  <->  E. y E. x  e.  A  y  =  ( rank `  x )
)
2422, 23sylibr 204 . . . . . 6  |-  ( A  =/=  (/)  ->  { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/) )
2511dfiin2 4086 . . . . . . 7  |-  |^|_ x  e.  A  ( rank `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( rank `  x ) }
26 rankon 7677 . . . . . . . . . . 11  |-  ( rank `  x )  e.  On
27 eleq1 2464 . . . . . . . . . . 11  |-  ( y  =  ( rank `  x
)  ->  ( y  e.  On  <->  ( rank `  x
)  e.  On ) )
2826, 27mpbiri 225 . . . . . . . . . 10  |-  ( y  =  ( rank `  x
)  ->  y  e.  On )
2928rexlimivw 2786 . . . . . . . . 9  |-  ( E. x  e.  A  y  =  ( rank `  x
)  ->  y  e.  On )
3029abssi 3378 . . . . . . . 8  |-  { y  |  E. x  e.  A  y  =  (
rank `  x ) }  C_  On
31 onint 4734 . . . . . . . 8  |-  ( ( { 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 652 . . . . . . 7  |-  ( { 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 2488 . . . . . 6  |-  ( { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/)  ->  |^|_ x  e.  A  ( rank `  x
)  e.  { y  |  E. x  e.  A  y  =  (
rank `  x ) } )
3424, 33syl 16 . . . . 5  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( rank `  x
)  e.  { y  |  E. x  e.  A  y  =  (
rank `  x ) } )
35 nfii1 4082 . . . . . . . . 9  |-  F/_ x |^|_ x  e.  A  (
rank `  x )
3635nfeq2 2551 . . . . . . . 8  |-  F/ x  y  =  |^|_ x  e.  A  ( rank `  x
)
37 eqeq1 2410 . . . . . . . 8  |-  ( y  =  |^|_ x  e.  A  ( rank `  x )  ->  ( y  =  (
rank `  x )  <->  |^|_
x  e.  A  (
rank `  x )  =  ( rank `  x
) ) )
3836, 37rexbid 2685 . . . . . . 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 )
) )
3938elabg 3043 . . . . . 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
) ) )
4039ibi 233 . . . . 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
) )
41 ssid 3327 . . . . . . . . . 10  |-  ( rank `  y )  C_  ( rank `  y )
42 fveq2 5687 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( rank `  x )  =  ( rank `  y
) )
4342sseq1d 3335 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  y
)  C_  ( rank `  y ) ) )
4443rspcev 3012 . . . . . . . . . 10  |-  ( ( y  e.  A  /\  ( rank `  y )  C_  ( rank `  y
) )  ->  E. x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
4541, 44mpan2 653 . . . . . . . . 9  |-  ( y  e.  A  ->  E. x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
46 iinss 4102 . . . . . . . . 9  |-  ( E. x  e.  A  (
rank `  x )  C_  ( rank `  y
)  ->  |^|_ x  e.  A  ( rank `  x
)  C_  ( rank `  y ) )
4745, 46syl 16 . . . . . . . 8  |-  ( y  e.  A  ->  |^|_ x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
48 sseq1 3329 . . . . . . . 8  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  ( |^|_ x  e.  A  ( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
4947, 48syl5ib 211 . . . . . . 7  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  ( y  e.  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
5049ralrimiv 2748 . . . . . 6  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) )
5150reximi 2773 . . . . 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
) )
5234, 40, 513syl 19 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) )
53 rabn0 3607 . . . 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 ) )
5452, 53sylibr 204 . . 3  |-  ( A  =/=  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =/=  (/) )
5554necon4i 2627 . 2  |-  ( { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/)  ->  A  =  (/) )
563, 55impbii 181 1  |-  ( A  =  (/)  <->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   (/)c0 3588   |^|cint 4010   |^|_ciin 4054   Oncon0 4541   ` cfv 5413   rankcrnk 7645
This theorem is referenced by:  scott0s  7768  cplem1  7769  karden  7775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646  df-rank 7647
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