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Theorem inttsk 8941
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 753 . . . . . . . 8  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A  C_  Tarski )
21sselda 3356 . . . . . . 7  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  t  e.  Tarski )
3 elinti 4137 . . . . . . . . 9  |-  ( z  e.  |^| A  ->  (
t  e.  A  -> 
z  e.  t ) )
43imp 429 . . . . . . . 8  |-  ( ( z  e.  |^| A  /\  t  e.  A
)  ->  z  e.  t )
54adantll 713 . . . . . . 7  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  z  e.  t )
6 tskpwss 8919 . . . . . . 7  |-  ( ( t  e.  Tarski  /\  z  e.  t )  ->  ~P z  C_  t )
72, 5, 6syl2anc 661 . . . . . 6  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  ~P z  C_  t )
87ralrimiva 2799 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A. t  e.  A  ~P z  C_  t )
9 ssint 4144 . . . . 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 8920 . . . . . . 7  |-  ( ( t  e.  Tarski  /\  z  e.  t )  ->  ~P z  e.  t )
122, 5, 11syl2anc 661 . . . . . 6  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  ~P z  e.  t )
1312ralrimiva 2799 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A. t  e.  A  ~P z  e.  t
)
14 vex 2975 . . . . . . 7  |-  z  e. 
_V
1514pwex 4475 . . . . . 6  |-  ~P z  e.  _V
1615elint2 4135 . . . . 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 532 . . 3  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  -> 
( ~P z  C_  |^| A  /\  ~P z  e.  |^| A ) )
1918ralrimiva 2799 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A. z  e.  |^| A ( ~P z  C_  |^| A  /\  ~P z  e.  |^| A
) )
20 elpwi 3869 . . . 4  |-  ( z  e.  ~P |^| A  ->  z  C_  |^| A )
21 rexnal 2726 . . . . . . . 8  |-  ( E. t  e.  A  -.  z  e.  t  <->  -.  A. t  e.  A  z  e.  t )
22 simpr 461 . . . . . . . . . . . . 13  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A  =/=  (/) )
23 intex 4448 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
2422, 23sylib 196 . . . . . . . . . . . 12  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
2524ad2antrr 725 . . . . . . . . . . 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 7355 . . . . . . . . . . 11  |-  ( |^| A  e.  _V  ->  ( z  C_  |^| A  -> 
z  ~<_  |^| A ) )
2825, 26, 27sylc 60 . . . . . . . . . 10  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~<_  |^| A )
29 vex 2975 . . . . . . . . . . . 12  |-  t  e. 
_V
30 intss1 4143 . . . . . . . . . . . . 13  |-  ( t  e.  A  ->  |^| A  C_  t )
3130ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  C_  t )
32 ssdomg 7355 . . . . . . . . . . . 12  |-  ( t  e.  _V  ->  ( |^| A  C_  t  ->  |^| A  ~<_  t ) )
3329, 31, 32mpsyl 63 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  ~<_  t )
34 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  -.  z  e.  t )
35 simplll 757 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  A  C_ 
Tarski )
36 simprl 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  e.  A )
3735, 36sseldd 3357 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  e.  Tarski )
3826, 31sstrd 3366 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  C_  t )
39 tsken 8921 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  Tarski  /\  z  C_  t )  ->  (
z  ~~  t  \/  z  e.  t )
)
4037, 38, 39syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  (
z  ~~  t  \/  z  e.  t )
)
4140ord 377 . . . . . . . . . . . . 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 7360 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  ~~  z )
44 domentr 7368 . . . . . . . . . . 11  |-  ( (
|^| A  ~<_  t  /\  t  ~~  z )  ->  |^| A  ~<_  z )
4533, 43, 44syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  ~<_  z )
46 sbth 7431 . . . . . . . . . 10  |-  ( ( z  ~<_  |^| A  /\  |^| A  ~<_  z )  -> 
z  ~~  |^| A )
4728, 45, 46syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~~  |^| A )
4847rexlimdvaa 2842 . . . . . . . 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 4135 . . . . . 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 378 . . . 4  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( z  ~~  |^| A  \/  z  e.  |^| A ) )
5420, 53sylan2 474 . . 3  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  ~P |^| A )  ->  ( z  ~~  |^| A  \/  z  e. 
|^| A ) )
5554ralrimiva 2799 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A. z  e.  ~P  |^| A ( z 
~~  |^| A  \/  z  e.  |^| A ) )
56 eltsk2g 8918 . . 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 16 . 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 913 1  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  Tarski )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   _Vcvv 2972    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   |^|cint 4128   class class class wbr 4292    ~~ cen 7307    ~<_ cdom 7308   Tarskictsk 8915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-er 7101  df-en 7311  df-dom 7312  df-tsk 8916
This theorem is referenced by:  tskmcl  9008
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