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Theorem scott0 7551
Description: Scott's trick collects all sets that have a certain property and are of 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 2783 . . 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 3476 . . 3  |-  { x  e.  (/)  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/)
31, 2syl6eq 2332 . 2  |-  ( A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/) )
4 n0 3465 . . . . . . . . 9  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 nfre1 2600 . . . . . . . . . 10  |-  F/ x E. x  e.  A  ( rank `  x )  =  ( rank `  x
)
6 eqid 2284 . . . . . . . . . . 11  |-  ( rank `  x )  =  (
rank `  x )
7 rspe 2605 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) )  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
86, 7mpan2 655 . . . . . . . . . 10  |-  ( x  e.  A  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
95, 8exlimi 1805 . . . . . . . . 9  |-  ( E. x  x  e.  A  ->  E. x  e.  A  ( rank `  x )  =  ( rank `  x
) )
104, 9sylbi 189 . . . . . . . 8  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( rank `  x )  =  (
rank `  x )
)
11 fvex 5499 . . . . . . . . . . . 12  |-  ( rank `  x )  e.  _V
12 eqeq1 2290 . . . . . . . . . . . . 13  |-  ( y  =  ( rank `  x
)  ->  ( y  =  ( rank `  x
)  <->  ( rank `  x
)  =  ( rank `  x ) ) )
1312anbi2d 687 . . . . . . . . . . . 12  |-  ( y  =  ( rank `  x
)  ->  ( (
x  e.  A  /\  y  =  ( rank `  x ) )  <->  ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) ) ) )
1411, 13spcev 2876 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) )  ->  E. y
( x  e.  A  /\  y  =  ( rank `  x ) ) )
1514eximi 1568 . . . . . . . . . 10  |-  ( E. x ( x  e.  A  /\  ( rank `  x )  =  (
rank `  x )
)  ->  E. x E. y ( x  e.  A  /\  y  =  ( rank `  x
) ) )
16 excom 1790 . . . . . . . . . 10  |-  ( E. y E. x ( x  e.  A  /\  y  =  ( rank `  x ) )  <->  E. x E. y ( x  e.  A  /\  y  =  ( rank `  x
) ) )
1715, 16sylibr 205 . . . . . . . . 9  |-  ( E. x ( x  e.  A  /\  ( rank `  x )  =  (
rank `  x )
)  ->  E. y E. x ( x  e.  A  /\  y  =  ( rank `  x
) ) )
18 df-rex 2550 . . . . . . . . 9  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  x
)  <->  E. x ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) ) )
19 df-rex 2550 . . . . . . . . . 10  |-  ( E. x  e.  A  y  =  ( rank `  x
)  <->  E. x ( x  e.  A  /\  y  =  ( rank `  x
) ) )
2019exbii 1574 . . . . . . . . 9  |-  ( E. y E. x  e.  A  y  =  (
rank `  x )  <->  E. y E. x ( x  e.  A  /\  y  =  ( rank `  x ) ) )
2117, 18, 203imtr4i 259 . . . . . . . 8  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  x
)  ->  E. y E. x  e.  A  y  =  ( rank `  x ) )
2210, 21syl 17 . . . . . . 7  |-  ( A  =/=  (/)  ->  E. y E. x  e.  A  y  =  ( rank `  x ) )
23 abn0 3474 . . . . . . 7  |-  ( { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/)  <->  E. y E. x  e.  A  y  =  ( rank `  x )
)
2422, 23sylibr 205 . . . . . 6  |-  ( A  =/=  (/)  ->  { y  |  E. x  e.  A  y  =  ( rank `  x ) }  =/=  (/) )
2511dfiin2 3939 . . . . . . 7  |-  |^|_ x  e.  A  ( rank `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( rank `  x ) }
26 rankon 7462 . . . . . . . . . . 11  |-  ( rank `  x )  e.  On
27 eleq1 2344 . . . . . . . . . . 11  |-  ( y  =  ( rank `  x
)  ->  ( y  e.  On  <->  ( rank `  x
)  e.  On ) )
2826, 27mpbiri 226 . . . . . . . . . 10  |-  ( y  =  ( rank `  x
)  ->  y  e.  On )
2928rexlimivw 2664 . . . . . . . . 9  |-  ( E. x  e.  A  y  =  ( rank `  x
)  ->  y  e.  On )
3029abssi 3249 . . . . . . . 8  |-  { y  |  E. x  e.  A  y  =  (
rank `  x ) }  C_  On
31 onint 4585 . . . . . . . 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 654 . . . . . . 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 2368 . . . . . 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 17 . . . . 5  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( rank `  x
)  e.  { y  |  E. x  e.  A  y  =  (
rank `  x ) } )
35 nfii1 3935 . . . . . . . . 9  |-  F/_ x |^|_ x  e.  A  (
rank `  x )
3635nfeq2 2431 . . . . . . . 8  |-  F/ x  y  =  |^|_ x  e.  A  ( rank `  x
)
37 eqeq1 2290 . . . . . . . 8  |-  ( y  =  |^|_ x  e.  A  ( rank `  x )  ->  ( y  =  (
rank `  x )  <->  |^|_
x  e.  A  (
rank `  x )  =  ( rank `  x
) ) )
3836, 37rexbid 2563 . . . . . . 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 2916 . . . . . 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 234 . . . . 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 3198 . . . . . . . . . 10  |-  ( rank `  y )  C_  ( rank `  y )
42 fveq2 5485 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( rank `  x )  =  ( rank `  y
) )
4342sseq1d 3206 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  y
)  C_  ( rank `  y ) ) )
4443rspcev 2885 . . . . . . . . . 10  |-  ( ( y  e.  A  /\  ( rank `  y )  C_  ( rank `  y
) )  ->  E. x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
4541, 44mpan2 655 . . . . . . . . 9  |-  ( y  e.  A  ->  E. x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
46 iinss 3954 . . . . . . . . 9  |-  ( E. x  e.  A  (
rank `  x )  C_  ( rank `  y
)  ->  |^|_ x  e.  A  ( rank `  x
)  C_  ( rank `  y ) )
4745, 46syl 17 . . . . . . . 8  |-  ( y  e.  A  ->  |^|_ x  e.  A  ( rank `  x )  C_  ( rank `  y ) )
48 sseq1 3200 . . . . . . . 8  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  ( |^|_ x  e.  A  ( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
4947, 48syl5ib 212 . . . . . . 7  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  ( y  e.  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
5049ralrimiv 2626 . . . . . 6  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) )
5150reximi 2651 . . . . 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 20 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) )
53 rabn0 3475 . . . 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 205 . . 3  |-  ( A  =/=  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =/=  (/) )
5554necon4i 2507 . 2  |-  ( { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/)  ->  A  =  (/) )
563, 55impbii 182 1  |-  ( A  =  (/)  <->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1533    = wceq 1628    e. wcel 1688   {cab 2270    =/= wne 2447   A.wral 2544   E.wrex 2545   {crab 2548    C_ wss 3153   (/)c0 3456   |^|cint 3863   |^|_ciin 3907   Oncon0 4391   ` cfv 5221   rankcrnk 7430
This theorem is referenced by:  scott0s  7553  cplem1  7554  karden  7560
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6383  df-rdg 6418  df-r1 7431  df-rank 7432
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