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Theorem scott0 8293
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 3100 . . 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 3799 . . 3  |-  { x  e.  (/)  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/)
31, 2syl6eq 2517 . 2  |-  ( A  =  (/)  ->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/) )
4 n0 3787 . . . . . . . 8  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 nfre1 2918 . . . . . . . . 9  |-  F/ x E. x  e.  A  ( rank `  x )  =  ( rank `  x
)
6 eqid 2460 . . . . . . . . . 10  |-  ( rank `  x )  =  (
rank `  x )
7 rspe 2915 . . . . . . . . . 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 1854 . . . . . . . 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 5867 . . . . . . . . . . 11  |-  ( rank `  x )  e.  _V
12 eqeq1 2464 . . . . . . . . . . . 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 3198 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) )  ->  E. y
( x  e.  A  /\  y  =  ( rank `  x ) ) )
1514eximi 1630 . . . . . . . . 9  |-  ( E. x ( x  e.  A  /\  ( rank `  x )  =  (
rank `  x )
)  ->  E. x E. y ( x  e.  A  /\  y  =  ( rank `  x
) ) )
16 excom 1793 . . . . . . . . 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 2813 . . . . . . . 8  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  x
)  <->  E. x ( x  e.  A  /\  ( rank `  x )  =  ( rank `  x
) ) )
19 df-rex 2813 . . . . . . . . 9  |-  ( E. x  e.  A  y  =  ( rank `  x
)  <->  E. x ( x  e.  A  /\  y  =  ( rank `  x
) ) )
2019exbii 1639 . . . . . . . 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 3797 . . . . . 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 4353 . . . . . 6  |-  |^|_ x  e.  A  ( rank `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( rank `  x ) }
26 rankon 8202 . . . . . . . . . 10  |-  ( rank `  x )  e.  On
27 eleq1 2532 . . . . . . . . . 10  |-  ( y  =  ( rank `  x
)  ->  ( y  e.  On  <->  ( rank `  x
)  e.  On ) )
2826, 27mpbiri 233 . . . . . . . . 9  |-  ( y  =  ( rank `  x
)  ->  y  e.  On )
2928rexlimivw 2945 . . . . . . . 8  |-  ( E. x  e.  A  y  =  ( rank `  x
)  ->  y  e.  On )
3029abssi 3568 . . . . . . 7  |-  { y  |  E. x  e.  A  y  =  (
rank `  x ) }  C_  On
31 onint 6601 . . . . . . 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 2552 . . . . 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 4349 . . . . . . . . 9  |-  F/_ x |^|_ x  e.  A  (
rank `  x )
3534nfeq2 2639 . . . . . . . 8  |-  F/ x  y  =  |^|_ x  e.  A  ( rank `  x
)
36 eqeq1 2464 . . . . . . . 8  |-  ( y  =  |^|_ x  e.  A  ( rank `  x )  ->  ( y  =  (
rank `  x )  <->  |^|_
x  e.  A  (
rank `  x )  =  ( rank `  x
) ) )
3735, 36rexbid 2965 . . . . . . 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 3244 . . . . . 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 3516 . . . . . . . . . 10  |-  ( rank `  y )  C_  ( rank `  y )
41 fveq2 5857 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( rank `  x )  =  ( rank `  y
) )
4241sseq1d 3524 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  y
)  C_  ( rank `  y ) ) )
4342rspcev 3207 . . . . . . . . . 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 4369 . . . . . . . . 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 3518 . . . . . . . 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 2869 . . . . . 6  |-  ( |^|_ x  e.  A  ( rank `  x )  =  (
rank `  x )  ->  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) )
5049reximi 2925 . . . . 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 3798 . . . 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 2704 . 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 1374   E.wex 1591    e. wcel 1762   {cab 2445    =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811    C_ wss 3469   (/)c0 3778   |^|cint 4275   |^|_ciin 4319   Oncon0 4871   ` cfv 5579   rankcrnk 8170
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-recs 7032  df-rdg 7066  df-r1 8171  df-rank 8172
This theorem is referenced by:  scott0s  8295  cplem1  8296  karden  8302  scott0f  30168
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