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Theorem cmpfii 17136
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 5539 . . . . 5  |-  ( Clsd `  J )  e.  _V
21elpw2 4175 . . . 4  |-  ( X  e.  ~P ( Clsd `  J )  <->  X  C_  ( Clsd `  J ) )
32biimpri 197 . . 3  |-  ( X 
C_  ( Clsd `  J
)  ->  X  e.  ~P ( Clsd `  J
) )
4 cmptop 17122 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
5 cmpfi 17135 . . . . 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 5525 . . . . . . 7  |-  ( x  =  X  ->  ( fi `  x )  =  ( fi `  X
) )
98eleq2d 2350 . . . . . 6  |-  ( x  =  X  ->  ( (/) 
e.  ( fi `  x )  <->  (/)  e.  ( fi `  X ) ) )
109notbid 285 . . . . 5  |-  ( x  =  X  ->  ( -.  (/)  e.  ( fi
`  x )  <->  -.  (/)  e.  ( fi `  X ) ) )
11 inteq 3865 . . . . . 6  |-  ( x  =  X  ->  |^| x  =  |^| X )
1211neeq1d 2459 . . . . 5  |-  ( x  =  X  ->  ( |^| x  =/=  (/)  <->  |^| X  =/=  (/) ) )
1310, 12imbi12d 311 . . . 4  |-  ( x  =  X  ->  (
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) )  <->  ( -.  (/) 
e.  ( fi `  X )  ->  |^| X  =/=  (/) ) ) )
1413rspcva 2882 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   ` cfv 5255   ficfi 7164   Topctop 16631   Clsdccld 16753   Compccmp 17113
This theorem is referenced by:  fclscmpi  17724  cmpfiiin  26184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-top 16636  df-cld 16756  df-cmp 17114
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