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Theorem kur14lem9 23760
Description: Lemma for kur14 23762. Since the set  T is closed under closure and complement, it contains the minimal set  S as a subset, so  S also has at most  1 4 elements. (Indeed  S  =  T, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j  |-  J  e. 
Top
kur14lem.x  |-  X  = 
U. J
kur14lem.k  |-  K  =  ( cls `  J
)
kur14lem.i  |-  I  =  ( int `  J
)
kur14lem.a  |-  A  C_  X
kur14lem.b  |-  B  =  ( X  \  ( K `  A )
)
kur14lem.c  |-  C  =  ( K `  ( X  \  A ) )
kur14lem.d  |-  D  =  ( I `  ( K `  A )
)
kur14lem.t  |-  T  =  ( ( ( { A ,  ( X 
\  A ) ,  ( K `  A
) }  u.  { B ,  C , 
( I `  A
) } )  u. 
{ ( K `  B ) ,  D ,  ( K `  ( I `  A
) ) } )  u.  ( { ( I `  C ) ,  ( K `  D ) ,  ( I `  ( K `
 B ) ) }  u.  { ( K `  ( I `
 C ) ) ,  ( I `  ( K `  ( I `
 A ) ) ) } ) )
kur14lem.s  |-  S  = 
|^| { x  e.  ~P ~P X  |  ( A  e.  x  /\  A. y  e.  x  {
( X  \  y
) ,  ( K `
 y ) } 
C_  x ) }
Assertion
Ref Expression
kur14lem9  |-  ( S  e.  Fin  /\  ( # `
 S )  <_ ; 1 4 )
Distinct variable groups:    x, A    x, K    x, y, T   
x, X, y
Allowed substitution hints:    A( y)    B( x, y)    C( x, y)    D( x, y)    S( x, y)    I( x, y)    J( x, y)    K( y)

Proof of Theorem kur14lem9
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 kur14lem.s . . 3  |-  S  = 
|^| { x  e.  ~P ~P X  |  ( A  e.  x  /\  A. y  e.  x  {
( X  \  y
) ,  ( K `
 y ) } 
C_  x ) }
2 vex 2804 . . . . . 6  |-  s  e. 
_V
32elintrab 3890 . . . . 5  |-  ( s  e.  |^| { x  e. 
~P ~P X  | 
( A  e.  x  /\  A. y  e.  x  { ( X  \ 
y ) ,  ( K `  y ) }  C_  x ) } 
<-> 
A. x  e.  ~P  ~P X ( ( A  e.  x  /\  A. y  e.  x  {
( X  \  y
) ,  ( K `
 y ) } 
C_  x )  -> 
s  e.  x ) )
4 ssun1 3351 . . . . . . . 8  |-  { A ,  ( X  \  A ) ,  ( K `  A ) }  C_  ( { A ,  ( X  \  A ) ,  ( K `  A ) }  u.  { B ,  C ,  ( I `
 A ) } )
5 ssun1 3351 . . . . . . . . 9  |-  ( { A ,  ( X 
\  A ) ,  ( K `  A
) }  u.  { B ,  C , 
( I `  A
) } )  C_  ( ( { A ,  ( X  \  A ) ,  ( K `  A ) }  u.  { B ,  C ,  ( I `
 A ) } )  u.  { ( K `  B ) ,  D ,  ( K `  ( I `
 A ) ) } )
6 ssun1 3351 . . . . . . . . . 10  |-  ( ( { A ,  ( X  \  A ) ,  ( K `  A ) }  u.  { B ,  C , 
( I `  A
) } )  u. 
{ ( K `  B ) ,  D ,  ( K `  ( I `  A
) ) } ) 
C_  ( ( ( { A ,  ( X  \  A ) ,  ( K `  A ) }  u.  { B ,  C , 
( I `  A
) } )  u. 
{ ( K `  B ) ,  D ,  ( K `  ( I `  A
) ) } )  u.  ( { ( I `  C ) ,  ( K `  D ) ,  ( I `  ( K `
 B ) ) }  u.  { ( K `  ( I `
 C ) ) ,  ( I `  ( K `  ( I `
 A ) ) ) } ) )
7 kur14lem.t . . . . . . . . . 10  |-  T  =  ( ( ( { A ,  ( X 
\  A ) ,  ( K `  A
) }  u.  { B ,  C , 
( I `  A
) } )  u. 
{ ( K `  B ) ,  D ,  ( K `  ( I `  A
) ) } )  u.  ( { ( I `  C ) ,  ( K `  D ) ,  ( I `  ( K `
 B ) ) }  u.  { ( K `  ( I `
 C ) ) ,  ( I `  ( K `  ( I `
 A ) ) ) } ) )
86, 7sseqtr4i 3224 . . . . . . . . 9  |-  ( ( { A ,  ( X  \  A ) ,  ( K `  A ) }  u.  { B ,  C , 
( I `  A
) } )  u. 
{ ( K `  B ) ,  D ,  ( K `  ( I `  A
) ) } ) 
C_  T
95, 8sstri 3201 . . . . . . . 8  |-  ( { A ,  ( X 
\  A ) ,  ( K `  A
) }  u.  { B ,  C , 
( I `  A
) } )  C_  T
104, 9sstri 3201 . . . . . . 7  |-  { A ,  ( X  \  A ) ,  ( K `  A ) }  C_  T
11 kur14lem.j . . . . . . . . . . 11  |-  J  e. 
Top
12 kur14lem.x . . . . . . . . . . . 12  |-  X  = 
U. J
1312topopn 16668 . . . . . . . . . . 11  |-  ( J  e.  Top  ->  X  e.  J )
1411, 13ax-mp 8 . . . . . . . . . 10  |-  X  e.  J
1514elexi 2810 . . . . . . . . 9  |-  X  e. 
_V
16 kur14lem.a . . . . . . . . 9  |-  A  C_  X
1715, 16ssexi 4175 . . . . . . . 8  |-  A  e. 
_V
1817tpid1 3752 . . . . . . 7  |-  A  e. 
{ A ,  ( X  \  A ) ,  ( K `  A ) }
1910, 18sselii 3190 . . . . . 6  |-  A  e.  T
20 kur14lem.k . . . . . . . . 9  |-  K  =  ( cls `  J
)
21 kur14lem.i . . . . . . . . 9  |-  I  =  ( int `  J
)
22 kur14lem.b . . . . . . . . 9  |-  B  =  ( X  \  ( K `  A )
)
23 kur14lem.c . . . . . . . . 9  |-  C  =  ( K `  ( X  \  A ) )
24 kur14lem.d . . . . . . . . 9  |-  D  =  ( I `  ( K `  A )
)
2511, 12, 20, 21, 16, 22, 23, 24, 7kur14lem7 23758 . . . . . . . 8  |-  ( y  e.  T  ->  (
y  C_  X  /\  { ( X  \  y
) ,  ( K `
 y ) } 
C_  T ) )
2625simprd 449 . . . . . . 7  |-  ( y  e.  T  ->  { ( X  \  y ) ,  ( K `  y ) }  C_  T )
2726rgen 2621 . . . . . 6  |-  A. y  e.  T  { ( X  \  y ) ,  ( K `  y
) }  C_  T
2825simpld 445 . . . . . . . . . 10  |-  ( y  e.  T  ->  y  C_  X )
2915elpw2 4191 . . . . . . . . . 10  |-  ( y  e.  ~P X  <->  y  C_  X )
3028, 29sylibr 203 . . . . . . . . 9  |-  ( y  e.  T  ->  y  e.  ~P X )
3130ssriv 3197 . . . . . . . 8  |-  T  C_  ~P X
3215pwex 4209 . . . . . . . . 9  |-  ~P X  e.  _V
3332elpw2 4191 . . . . . . . 8  |-  ( T  e.  ~P ~P X  <->  T 
C_  ~P X )
3431, 33mpbir 200 . . . . . . 7  |-  T  e. 
~P ~P X
35 eleq2 2357 . . . . . . . . . 10  |-  ( x  =  T  ->  ( A  e.  x  <->  A  e.  T ) )
36 sseq2 3213 . . . . . . . . . . 11  |-  ( x  =  T  ->  ( { ( X  \ 
y ) ,  ( K `  y ) }  C_  x  <->  { ( X  \  y ) ,  ( K `  y
) }  C_  T
) )
3736raleqbi1dv 2757 . . . . . . . . . 10  |-  ( x  =  T  ->  ( A. y  e.  x  { ( X  \ 
y ) ,  ( K `  y ) }  C_  x  <->  A. y  e.  T  { ( X  \  y ) ,  ( K `  y
) }  C_  T
) )
3835, 37anbi12d 691 . . . . . . . . 9  |-  ( x  =  T  ->  (
( A  e.  x  /\  A. y  e.  x  { ( X  \ 
y ) ,  ( K `  y ) }  C_  x )  <->  ( A  e.  T  /\  A. y  e.  T  {
( X  \  y
) ,  ( K `
 y ) } 
C_  T ) ) )
39 eleq2 2357 . . . . . . . . 9  |-  ( x  =  T  ->  (
s  e.  x  <->  s  e.  T ) )
4038, 39imbi12d 311 . . . . . . . 8  |-  ( x  =  T  ->  (
( ( A  e.  x  /\  A. y  e.  x  { ( X  \  y ) ,  ( K `  y
) }  C_  x
)  ->  s  e.  x )  <->  ( ( A  e.  T  /\  A. y  e.  T  {
( X  \  y
) ,  ( K `
 y ) } 
C_  T )  -> 
s  e.  T ) ) )
4140rspccv 2894 . . . . . . 7  |-  ( A. x  e.  ~P  ~P X
( ( A  e.  x  /\  A. y  e.  x  { ( X  \  y ) ,  ( K `  y
) }  C_  x
)  ->  s  e.  x )  ->  ( T  e.  ~P ~P X  ->  ( ( A  e.  T  /\  A. y  e.  T  {
( X  \  y
) ,  ( K `
 y ) } 
C_  T )  -> 
s  e.  T ) ) )
4234, 41mpi 16 . . . . . 6  |-  ( A. x  e.  ~P  ~P X
( ( A  e.  x  /\  A. y  e.  x  { ( X  \  y ) ,  ( K `  y
) }  C_  x
)  ->  s  e.  x )  ->  (
( A  e.  T  /\  A. y  e.  T  { ( X  \ 
y ) ,  ( K `  y ) }  C_  T )  ->  s  e.  T ) )
4319, 27, 42mp2ani 659 . . . . 5  |-  ( A. x  e.  ~P  ~P X
( ( A  e.  x  /\  A. y  e.  x  { ( X  \  y ) ,  ( K `  y
) }  C_  x
)  ->  s  e.  x )  ->  s  e.  T )
443, 43sylbi 187 . . . 4  |-  ( s  e.  |^| { x  e. 
~P ~P X  | 
( A  e.  x  /\  A. y  e.  x  { ( X  \ 
y ) ,  ( K `  y ) }  C_  x ) }  ->  s  e.  T
)
4544ssriv 3197 . . 3  |-  |^| { x  e.  ~P ~P X  | 
( A  e.  x  /\  A. y  e.  x  { ( X  \ 
y ) ,  ( K `  y ) }  C_  x ) }  C_  T
461, 45eqsstri 3221 . 2  |-  S  C_  T
4711, 12, 20, 21, 16, 22, 23, 24, 7kur14lem8 23759 . 2  |-  ( T  e.  Fin  /\  ( # `
 T )  <_ ; 1 4 )
48 1nn0 9997 . . 3  |-  1  e.  NN0
49 4nn0 10000 . . 3  |-  4  e.  NN0
5048, 49deccl 10154 . 2  |- ; 1 4  e.  NN0
5146, 47, 50hashsslei 11394 1  |-  ( S  e.  Fin  /\  ( # `
 S )  <_ ; 1 4 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    u. cun 3163    C_ wss 3165   ~Pcpw 3638   {cpr 3654   {ctp 3655   U.cuni 3843   |^|cint 3878   class class class wbr 4039   ` cfv 5271   Fincfn 6879   1c1 8754    <_ cle 8884   4c4 9813  ;cdc 10140   #chash 11353   Topctop 16647   intcnt 16770   clsccl 16771
This theorem is referenced by:  kur14lem10  23761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-hash 11354  df-top 16652  df-cld 16772  df-ntr 16773  df-cls 16774
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