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Theorem cmpfii 19754
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 5881 . . . . 5  |-  ( Clsd `  J )  e.  _V
21elpw2 4616 . . . 4  |-  ( X  e.  ~P ( Clsd `  J )  <->  X  C_  ( Clsd `  J ) )
32biimpri 206 . . 3  |-  ( X 
C_  ( Clsd `  J
)  ->  X  e.  ~P ( Clsd `  J
) )
4 cmptop 19740 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
5 cmpfi 19753 . . . . 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 5871 . . . . . . 7  |-  ( x  =  X  ->  ( fi `  x )  =  ( fi `  X
) )
98eleq2d 2537 . . . . . 6  |-  ( x  =  X  ->  ( (/) 
e.  ( fi `  x )  <->  (/)  e.  ( fi `  X ) ) )
109notbid 294 . . . . 5  |-  ( x  =  X  ->  ( -.  (/)  e.  ( fi
`  x )  <->  -.  (/)  e.  ( fi `  X ) ) )
11 inteq 4290 . . . . . 6  |-  ( x  =  X  ->  |^| x  =  |^| X )
1211neeq1d 2744 . . . . 5  |-  ( x  =  X  ->  ( |^| x  =/=  (/)  <->  |^| X  =/=  (/) ) )
1310, 12imbi12d 320 . . . 4  |-  ( x  =  X  ->  (
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) )  <->  ( -.  (/) 
e.  ( fi `  X )  ->  |^| X  =/=  (/) ) ) )
1413rspcva 3217 . . 3  |-  ( ( X  e.  ~P ( Clsd `  J )  /\  A. x  e.  ~P  ( Clsd `  J ) ( -.  (/)  e.  ( fi
`  x )  ->  |^| x  =/=  (/) ) )  ->  ( -.  (/)  e.  ( fi `  X )  ->  |^| X  =/=  (/) ) )
153, 7, 14syl2anr 478 . 2  |-  ( ( J  e.  Comp  /\  X  C_  ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  X )  ->  |^| X  =/=  (/) ) )
16153impia 1193 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817    C_ wss 3481   (/)c0 3790   ~Pcpw 4015   |^|cint 4287   ` cfv 5593   ficfi 7880   Topctop 19240   Clsdccld 19362   Compccmp 19731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fi 7881  df-top 19245  df-cld 19365  df-cmp 19732
This theorem is referenced by:  fclscmpi  20375  cmpfiiin  30525
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