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Theorem inttsk 8937
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 748 . . . . . . . 8  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A  C_  Tarski )
21sselda 3353 . . . . . . 7  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  t  e.  Tarski )
3 elinti 4134 . . . . . . . . 9  |-  ( z  e.  |^| A  ->  (
t  e.  A  -> 
z  e.  t ) )
43imp 429 . . . . . . . 8  |-  ( ( z  e.  |^| A  /\  t  e.  A
)  ->  z  e.  t )
54adantll 708 . . . . . . 7  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  z  e.  t )
6 tskpwss 8915 . . . . . . 7  |-  ( ( t  e.  Tarski  /\  z  e.  t )  ->  ~P z  C_  t )
72, 5, 6syl2anc 656 . . . . . 6  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  ~P z  C_  t )
87ralrimiva 2797 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A. t  e.  A  ~P z  C_  t )
9 ssint 4141 . . . . 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 8916 . . . . . . 7  |-  ( ( t  e.  Tarski  /\  z  e.  t )  ->  ~P z  e.  t )
122, 5, 11syl2anc 656 . . . . . 6  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  ~P z  e.  t )
1312ralrimiva 2797 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A. t  e.  A  ~P z  e.  t
)
14 vex 2973 . . . . . . 7  |-  z  e. 
_V
1514pwex 4472 . . . . . 6  |-  ~P z  e.  _V
1615elint2 4132 . . . . 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 529 . . 3  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  -> 
( ~P z  C_  |^| A  /\  ~P z  e.  |^| A ) )
1918ralrimiva 2797 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A. z  e.  |^| A ( ~P z  C_  |^| A  /\  ~P z  e.  |^| A
) )
20 elpwi 3866 . . . 4  |-  ( z  e.  ~P |^| A  ->  z  C_  |^| A )
21 rexnal 2724 . . . . . . . 8  |-  ( E. t  e.  A  -.  z  e.  t  <->  -.  A. t  e.  A  z  e.  t )
22 simpr 458 . . . . . . . . . . . . 13  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A  =/=  (/) )
23 intex 4445 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
2422, 23sylib 196 . . . . . . . . . . . 12  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
2524ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  e.  _V )
26 simplr 749 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  C_ 
|^| A )
27 ssdomg 7351 . . . . . . . . . . 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 2973 . . . . . . . . . . . 12  |-  t  e. 
_V
30 intss1 4140 . . . . . . . . . . . . 13  |-  ( t  e.  A  ->  |^| A  C_  t )
3130ad2antrl 722 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  C_  t )
32 ssdomg 7351 . . . . . . . . . . . 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 751 . . . . . . . . . . . . 13  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  -.  z  e.  t )
35 simplll 752 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  A  C_ 
Tarski )
36 simprl 750 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  e.  A )
3735, 36sseldd 3354 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  e.  Tarski )
3826, 31sstrd 3363 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  C_  t )
39 tsken 8917 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  Tarski  /\  z  C_  t )  ->  (
z  ~~  t  \/  z  e.  t )
)
4037, 38, 39syl2anc 656 . . . . . . . . . . . . . 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 7356 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  ~~  z )
44 domentr 7364 . . . . . . . . . . 11  |-  ( (
|^| A  ~<_  t  /\  t  ~~  z )  ->  |^| A  ~<_  z )
4533, 43, 44syl2anc 656 . . . . . . . . . 10  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  ~<_  z )
46 sbth 7427 . . . . . . . . . 10  |-  ( ( z  ~<_  |^| A  /\  |^| A  ~<_  z )  -> 
z  ~~  |^| A )
4728, 45, 46syl2anc 656 . . . . . . . . 9  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~~  |^| A )
4847rexlimdvaa 2840 . . . . . . . 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 4132 . . . . . 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 471 . . 3  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  ~P |^| A )  ->  ( z  ~~  |^| A  \/  z  e. 
|^| A ) )
5554ralrimiva 2797 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A. z  e.  ~P  |^| A ( z 
~~  |^| A  \/  z  e.  |^| A ) )
56 eltsk2g 8914 . . 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 908 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 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   |^|cint 4125   class class class wbr 4289    ~~ cen 7303    ~<_ cdom 7304   Tarskictsk 8911
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-int 4126  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-er 7097  df-en 7307  df-dom 7308  df-tsk 8912
This theorem is referenced by:  tskmcl  9004
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