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Theorem cmpfii 19012
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 5701 . . . . 5  |-  ( Clsd `  J )  e.  _V
21elpw2 4456 . . . 4  |-  ( X  e.  ~P ( Clsd `  J )  <->  X  C_  ( Clsd `  J ) )
32biimpri 206 . . 3  |-  ( X 
C_  ( Clsd `  J
)  ->  X  e.  ~P ( Clsd `  J
) )
4 cmptop 18998 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
5 cmpfi 19011 . . . . 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 5691 . . . . . . 7  |-  ( x  =  X  ->  ( fi `  x )  =  ( fi `  X
) )
98eleq2d 2510 . . . . . 6  |-  ( x  =  X  ->  ( (/) 
e.  ( fi `  x )  <->  (/)  e.  ( fi `  X ) ) )
109notbid 294 . . . . 5  |-  ( x  =  X  ->  ( -.  (/)  e.  ( fi
`  x )  <->  -.  (/)  e.  ( fi `  X ) ) )
11 inteq 4131 . . . . . 6  |-  ( x  =  X  ->  |^| x  =  |^| X )
1211neeq1d 2621 . . . . 5  |-  ( x  =  X  ->  ( |^| x  =/=  (/)  <->  |^| X  =/=  (/) ) )
1310, 12imbi12d 320 . . . 4  |-  ( x  =  X  ->  (
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) )  <->  ( -.  (/) 
e.  ( fi `  X )  ->  |^| X  =/=  (/) ) ) )
1413rspcva 3071 . . 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 1184 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   |^|cint 4128   ` cfv 5418   ficfi 7660   Topctop 18498   Clsdccld 18620   Compccmp 18989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fi 7661  df-top 18503  df-cld 18623  df-cmp 18990
This theorem is referenced by:  fclscmpi  19602  cmpfiiin  29033
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