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Theorem inttsk 9182
Description: The intersection of a collection of Tarski classes is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
inttsk  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  Tarski )

Proof of Theorem inttsk
Dummy variables  t 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 752 . . . . . . . 8  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A  C_  Tarski )
21sselda 3442 . . . . . . 7  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  t  e.  Tarski )
3 elinti 4236 . . . . . . . . 9  |-  ( z  e.  |^| A  ->  (
t  e.  A  -> 
z  e.  t ) )
43imp 427 . . . . . . . 8  |-  ( ( z  e.  |^| A  /\  t  e.  A
)  ->  z  e.  t )
54adantll 712 . . . . . . 7  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  z  e.  t )
6 tskpwss 9160 . . . . . . 7  |-  ( ( t  e.  Tarski  /\  z  e.  t )  ->  ~P z  C_  t )
72, 5, 6syl2anc 659 . . . . . 6  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  ~P z  C_  t )
87ralrimiva 2818 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A. t  e.  A  ~P z  C_  t )
9 ssint 4243 . . . . 5  |-  ( ~P z  C_  |^| A  <->  A. t  e.  A  ~P z  C_  t )
108, 9sylibr 212 . . . 4  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  ~P z  C_  |^| A
)
11 tskpw 9161 . . . . . . 7  |-  ( ( t  e.  Tarski  /\  z  e.  t )  ->  ~P z  e.  t )
122, 5, 11syl2anc 659 . . . . . 6  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  ~P z  e.  t )
1312ralrimiva 2818 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A. t  e.  A  ~P z  e.  t
)
14 vex 3062 . . . . . . 7  |-  z  e. 
_V
1514pwex 4577 . . . . . 6  |-  ~P z  e.  _V
1615elint2 4234 . . . . 5  |-  ( ~P z  e.  |^| A  <->  A. t  e.  A  ~P z  e.  t )
1713, 16sylibr 212 . . . 4  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  ~P z  e.  |^| A
)
1810, 17jca 530 . . 3  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  -> 
( ~P z  C_  |^| A  /\  ~P z  e.  |^| A ) )
1918ralrimiva 2818 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A. z  e.  |^| A ( ~P z  C_  |^| A  /\  ~P z  e.  |^| A
) )
20 elpwi 3964 . . . 4  |-  ( z  e.  ~P |^| A  ->  z  C_  |^| A )
21 rexnal 2852 . . . . . . . 8  |-  ( E. t  e.  A  -.  z  e.  t  <->  -.  A. t  e.  A  z  e.  t )
22 simpr 459 . . . . . . . . . . . . 13  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A  =/=  (/) )
23 intex 4550 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
2422, 23sylib 196 . . . . . . . . . . . 12  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
2524ad2antrr 724 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  e.  _V )
26 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  C_ 
|^| A )
27 ssdomg 7599 . . . . . . . . . . 11  |-  ( |^| A  e.  _V  ->  ( z  C_  |^| A  -> 
z  ~<_  |^| A ) )
2825, 26, 27sylc 59 . . . . . . . . . 10  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~<_  |^| A )
29 vex 3062 . . . . . . . . . . . 12  |-  t  e. 
_V
30 intss1 4242 . . . . . . . . . . . . 13  |-  ( t  e.  A  ->  |^| A  C_  t )
3130ad2antrl 726 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  C_  t )
32 ssdomg 7599 . . . . . . . . . . . 12  |-  ( t  e.  _V  ->  ( |^| A  C_  t  ->  |^| A  ~<_  t ) )
3329, 31, 32mpsyl 62 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  ~<_  t )
34 simprr 758 . . . . . . . . . . . . 13  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  -.  z  e.  t )
35 simplll 760 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  A  C_ 
Tarski )
36 simprl 756 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  e.  A )
3735, 36sseldd 3443 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  e.  Tarski )
3826, 31sstrd 3452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  C_  t )
39 tsken 9162 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  Tarski  /\  z  C_  t )  ->  (
z  ~~  t  \/  z  e.  t )
)
4037, 38, 39syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  (
z  ~~  t  \/  z  e.  t )
)
4140ord 375 . . . . . . . . . . . . 13  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  ( -.  z  ~~  t  -> 
z  e.  t ) )
4234, 41mt3d 125 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~~  t )
4342ensymd 7604 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  ~~  z )
44 domentr 7612 . . . . . . . . . . 11  |-  ( (
|^| A  ~<_  t  /\  t  ~~  z )  ->  |^| A  ~<_  z )
4533, 43, 44syl2anc 659 . . . . . . . . . 10  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  ~<_  z )
46 sbth 7675 . . . . . . . . . 10  |-  ( ( z  ~<_  |^| A  /\  |^| A  ~<_  z )  -> 
z  ~~  |^| A )
4728, 45, 46syl2anc 659 . . . . . . . . 9  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~~  |^| A )
4847rexlimdvaa 2897 . . . . . . . 8  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( E. t  e.  A  -.  z  e.  t  ->  z  ~~  |^| A ) )
4921, 48syl5bir 218 . . . . . . 7  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( -.  A. t  e.  A  z  e.  t  ->  z  ~~  |^| A ) )
5049con1d 124 . . . . . 6  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( -.  z  ~~  |^| A  ->  A. t  e.  A  z  e.  t ) )
5114elint2 4234 . . . . . 6  |-  ( z  e.  |^| A  <->  A. t  e.  A  z  e.  t )
5250, 51syl6ibr 227 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( -.  z  ~~  |^| A  ->  z  e.  |^| A ) )
5352orrd 376 . . . 4  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( z  ~~  |^| A  \/  z  e.  |^| A ) )
5420, 53sylan2 472 . . 3  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  ~P |^| A )  ->  ( z  ~~  |^| A  \/  z  e. 
|^| A ) )
5554ralrimiva 2818 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A. z  e.  ~P  |^| A ( z 
~~  |^| A  \/  z  e.  |^| A ) )
56 eltsk2g 9159 . . 3  |-  ( |^| A  e.  _V  ->  (
|^| A  e.  Tarski  <->  ( A. z  e.  |^| A
( ~P z  C_  |^| A  /\  ~P z  e.  |^| A )  /\  A. z  e.  ~P  |^| A ( z  ~~  |^| A  \/  z  e. 
|^| A ) ) ) )
5724, 56syl 17 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  ( |^| A  e.  Tarski  <->  ( A. z  e.  |^| A ( ~P z  C_  |^| A  /\  ~P z  e.  |^| A )  /\  A. z  e.  ~P  |^| A
( z  ~~  |^| A  \/  z  e.  |^| A ) ) ) )
5819, 55, 57mpbir2and 923 1  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  Tarski )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   _Vcvv 3059    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   |^|cint 4227   class class class wbr 4395    ~~ cen 7551    ~<_ cdom 7552   Tarskictsk 9156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-int 4228  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-er 7348  df-en 7555  df-dom 7556  df-tsk 9157
This theorem is referenced by:  tskmcl  9249
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