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Theorem cmpfii 18856
Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
cmpfii  |-  ( ( J  e.  Comp  /\  X  C_  ( Clsd `  J
)  /\  -.  (/)  e.  ( fi `  X ) )  ->  |^| X  =/=  (/) )

Proof of Theorem cmpfii
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvex 5691 . . . . 5  |-  ( Clsd `  J )  e.  _V
21elpw2 4446 . . . 4  |-  ( X  e.  ~P ( Clsd `  J )  <->  X  C_  ( Clsd `  J ) )
32biimpri 206 . . 3  |-  ( X 
C_  ( Clsd `  J
)  ->  X  e.  ~P ( Clsd `  J
) )
4 cmptop 18842 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
5 cmpfi 18855 . . . . 5  |-  ( J  e.  Top  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J
) ( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
64, 5syl 16 . . . 4  |-  ( J  e.  Comp  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J )
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
76ibi 241 . . 3  |-  ( J  e.  Comp  ->  A. x  e.  ~P  ( Clsd `  J
) ( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) )
8 fveq2 5681 . . . . . . 7  |-  ( x  =  X  ->  ( fi `  x )  =  ( fi `  X
) )
98eleq2d 2502 . . . . . 6  |-  ( x  =  X  ->  ( (/) 
e.  ( fi `  x )  <->  (/)  e.  ( fi `  X ) ) )
109notbid 294 . . . . 5  |-  ( x  =  X  ->  ( -.  (/)  e.  ( fi
`  x )  <->  -.  (/)  e.  ( fi `  X ) ) )
11 inteq 4121 . . . . . 6  |-  ( x  =  X  ->  |^| x  =  |^| X )
1211neeq1d 2613 . . . . 5  |-  ( x  =  X  ->  ( |^| x  =/=  (/)  <->  |^| X  =/=  (/) ) )
1310, 12imbi12d 320 . . . 4  |-  ( x  =  X  ->  (
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) )  <->  ( -.  (/) 
e.  ( fi `  X )  ->  |^| X  =/=  (/) ) ) )
1413rspcva 3062 . . 3  |-  ( ( X  e.  ~P ( Clsd `  J )  /\  A. x  e.  ~P  ( Clsd `  J ) ( -.  (/)  e.  ( fi
`  x )  ->  |^| x  =/=  (/) ) )  ->  ( -.  (/)  e.  ( fi `  X )  ->  |^| X  =/=  (/) ) )
153, 7, 14syl2anr 475 . 2  |-  ( ( J  e.  Comp  /\  X  C_  ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  X )  ->  |^| X  =/=  (/) ) )
16153impia 1179 1  |-  ( ( J  e.  Comp  /\  X  C_  ( Clsd `  J
)  /\  -.  (/)  e.  ( fi `  X ) )  ->  |^| X  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 960    = wceq 1364    e. wcel 1757    =/= wne 2598   A.wral 2707    C_ wss 3318   (/)c0 3627   ~Pcpw 3850   |^|cint 4118   ` cfv 5408   ficfi 7650   Topctop 18342   Clsdccld 18464   Compccmp 18833
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 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-iin 4164  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-om 6468  df-1st 6568  df-2nd 6569  df-recs 6820  df-rdg 6854  df-1o 6910  df-2o 6911  df-oadd 6914  df-er 7091  df-map 7206  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-fi 7651  df-top 18347  df-cld 18467  df-cmp 18834
This theorem is referenced by:  fclscmpi  19446  cmpfiiin  28880
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