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Theorem inttsk 8441
Description: The intersection of a collection of Tarski's classes is a Tarski's 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 730 . . . . . . . 8  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A  C_  Tarski )
21sselda 3214 . . . . . . 7  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  t  e.  Tarski )
3 elinti 3908 . . . . . . . . 9  |-  ( z  e.  |^| A  ->  (
t  e.  A  -> 
z  e.  t ) )
43imp 418 . . . . . . . 8  |-  ( ( z  e.  |^| A  /\  t  e.  A
)  ->  z  e.  t )
54adantll 694 . . . . . . 7  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  z  e.  t )
6 tskpwss 8419 . . . . . . 7  |-  ( ( t  e.  Tarski  /\  z  e.  t )  ->  ~P z  C_  t )
72, 5, 6syl2anc 642 . . . . . 6  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  ~P z  C_  t )
87ralrimiva 2660 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A. t  e.  A  ~P z  C_  t )
9 ssint 3915 . . . . 5  |-  ( ~P z  C_  |^| A  <->  A. t  e.  A  ~P z  C_  t )
108, 9sylibr 203 . . . 4  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  ~P z  C_  |^| A
)
11 tskpw 8420 . . . . . . 7  |-  ( ( t  e.  Tarski  /\  z  e.  t )  ->  ~P z  e.  t )
122, 5, 11syl2anc 642 . . . . . 6  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  /\  t  e.  A
)  ->  ~P z  e.  t )
1312ralrimiva 2660 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  A. t  e.  A  ~P z  e.  t
)
14 vex 2825 . . . . . . 7  |-  z  e. 
_V
1514pwex 4230 . . . . . 6  |-  ~P z  e.  _V
1615elint2 3906 . . . . 5  |-  ( ~P z  e.  |^| A  <->  A. t  e.  A  ~P z  e.  t )
1713, 16sylibr 203 . . . 4  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  ->  ~P z  e.  |^| A
)
1810, 17jca 518 . . 3  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  |^| A )  -> 
( ~P z  C_  |^| A  /\  ~P z  e.  |^| A ) )
1918ralrimiva 2660 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A. z  e.  |^| A ( ~P z  C_  |^| A  /\  ~P z  e.  |^| A
) )
20 elpwi 3667 . . . 4  |-  ( z  e.  ~P |^| A  ->  z  C_  |^| A )
21 rexnal 2588 . . . . . . . 8  |-  ( E. t  e.  A  -.  z  e.  t  <->  -.  A. t  e.  A  z  e.  t )
22 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A  =/=  (/) )
23 intex 4204 . . . . . . . . . . . . . 14  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
2422, 23sylib 188 . . . . . . . . . . . . 13  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
2524ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  e.  _V )
26 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  C_ 
|^| A )
27 ssdomg 6950 . . . . . . . . . . . 12  |-  ( |^| A  e.  _V  ->  ( z  C_  |^| A  -> 
z  ~<_  |^| A ) )
2825, 26, 27sylc 56 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~<_  |^| A )
29 vex 2825 . . . . . . . . . . . . 13  |-  t  e. 
_V
30 intss1 3914 . . . . . . . . . . . . . 14  |-  ( t  e.  A  ->  |^| A  C_  t )
3130ad2antrl 708 . . . . . . . . . . . . 13  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  C_  t )
32 ssdomg 6950 . . . . . . . . . . . . 13  |-  ( t  e.  _V  ->  ( |^| A  C_  t  ->  |^| A  ~<_  t ) )
3329, 31, 32mpsyl 59 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  ~<_  t )
34 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  -.  z  e.  t )
35 simplll 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  A  C_ 
Tarski )
36 simprl 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  e.  A )
3735, 36sseldd 3215 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  e.  Tarski )
3826, 31sstrd 3223 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  C_  t )
39 tsken 8421 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  Tarski  /\  z  C_  t )  ->  (
z  ~~  t  \/  z  e.  t )
)
4037, 38, 39syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  (
z  ~~  t  \/  z  e.  t )
)
4140ord 366 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  ( -.  z  ~~  t  -> 
z  e.  t ) )
4234, 41mt3d 117 . . . . . . . . . . . . 13  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~~  t )
43 ensym 6953 . . . . . . . . . . . . 13  |-  ( z 
~~  t  ->  t  ~~  z )
4442, 43syl 15 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  t  ~~  z )
45 domentr 6963 . . . . . . . . . . . 12  |-  ( (
|^| A  ~<_  t  /\  t  ~~  z )  ->  |^| A  ~<_  z )
4633, 44, 45syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  |^| A  ~<_  z )
47 sbth 7024 . . . . . . . . . . 11  |-  ( ( z  ~<_  |^| A  /\  |^| A  ~<_  z )  -> 
z  ~~  |^| A )
4828, 46, 47syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  ( t  e.  A  /\  -.  z  e.  t ) )  ->  z  ~~  |^| A )
4948expr 598 . . . . . . . . 9  |-  ( ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_  |^| A )  /\  t  e.  A )  ->  ( -.  z  e.  t  ->  z  ~~  |^| A ) )
5049rexlimdva 2701 . . . . . . . 8  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( E. t  e.  A  -.  z  e.  t  ->  z  ~~  |^| A ) )
5121, 50syl5bir 209 . . . . . . 7  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( -.  A. t  e.  A  z  e.  t  ->  z  ~~  |^| A ) )
5251con1d 116 . . . . . 6  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( -.  z  ~~  |^| A  ->  A. t  e.  A  z  e.  t ) )
5314elint2 3906 . . . . . 6  |-  ( z  e.  |^| A  <->  A. t  e.  A  z  e.  t )
5452, 53syl6ibr 218 . . . . 5  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( -.  z  ~~  |^| A  ->  z  e.  |^| A ) )
5554orrd 367 . . . 4  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  C_ 
|^| A )  -> 
( z  ~~  |^| A  \/  z  e.  |^| A ) )
5620, 55sylan2 460 . . 3  |-  ( ( ( A  C_  Tarski  /\  A  =/=  (/) )  /\  z  e.  ~P |^| A )  ->  ( z  ~~  |^| A  \/  z  e. 
|^| A ) )
5756ralrimiva 2660 . 2  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  A. z  e.  ~P  |^| A ( z 
~~  |^| A  \/  z  e.  |^| A ) )
58 eltsk2g 8418 . . 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 ) ) ) )
5924, 58syl 15 . 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 ) ) ) )
6019, 57, 59mpbir2and 888 1  |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  Tarski )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    e. wcel 1701    =/= wne 2479   A.wral 2577   E.wrex 2578   _Vcvv 2822    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   |^|cint 3899   class class class wbr 4060    ~~ cen 6903    ~<_ cdom 6904   Tarskictsk 8415
This theorem is referenced by:  tskmcl  8508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-er 6702  df-en 6907  df-dom 6908  df-tsk 8416
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