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Theorem cmpfii 17152
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 5555 . . . . 5  |-  ( Clsd `  J )  e.  _V
21elpw2 4191 . . . 4  |-  ( X  e.  ~P ( Clsd `  J )  <->  X  C_  ( Clsd `  J ) )
32biimpri 197 . . 3  |-  ( X 
C_  ( Clsd `  J
)  ->  X  e.  ~P ( Clsd `  J
) )
4 cmptop 17138 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
5 cmpfi 17151 . . . . 5  |-  ( J  e.  Top  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J
) ( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
64, 5syl 15 . . . 4  |-  ( J  e.  Comp  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J )
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
76ibi 232 . . 3  |-  ( J  e.  Comp  ->  A. x  e.  ~P  ( Clsd `  J
) ( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) )
8 fveq2 5541 . . . . . . 7  |-  ( x  =  X  ->  ( fi `  x )  =  ( fi `  X
) )
98eleq2d 2363 . . . . . 6  |-  ( x  =  X  ->  ( (/) 
e.  ( fi `  x )  <->  (/)  e.  ( fi `  X ) ) )
109notbid 285 . . . . 5  |-  ( x  =  X  ->  ( -.  (/)  e.  ( fi
`  x )  <->  -.  (/)  e.  ( fi `  X ) ) )
11 inteq 3881 . . . . . 6  |-  ( x  =  X  ->  |^| x  =  |^| X )
1211neeq1d 2472 . . . . 5  |-  ( x  =  X  ->  ( |^| x  =/=  (/)  <->  |^| X  =/=  (/) ) )
1310, 12imbi12d 311 . . . 4  |-  ( x  =  X  ->  (
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) )  <->  ( -.  (/) 
e.  ( fi `  X )  ->  |^| X  =/=  (/) ) ) )
1413rspcva 2895 . . 3  |-  ( ( X  e.  ~P ( Clsd `  J )  /\  A. x  e.  ~P  ( Clsd `  J ) ( -.  (/)  e.  ( fi
`  x )  ->  |^| x  =/=  (/) ) )  ->  ( -.  (/)  e.  ( fi `  X )  ->  |^| X  =/=  (/) ) )
153, 7, 14syl2anr 464 . 2  |-  ( ( J  e.  Comp  /\  X  C_  ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  X )  ->  |^| X  =/=  (/) ) )
16153impia 1148 1  |-  ( ( J  e.  Comp  /\  X  C_  ( Clsd `  J
)  /\  -.  (/)  e.  ( fi `  X ) )  ->  |^| X  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   ` cfv 5271   ficfi 7180   Topctop 16647   Clsdccld 16769   Compccmp 17129
This theorem is referenced by:  fclscmpi  17740  cmpfiiin  26875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-top 16652  df-cld 16772  df-cmp 17130
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