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Theorem filintn0 17550
Description: A filter has the finite intersection property. Remark below definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filintn0  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  =/=  (/) )

Proof of Theorem filintn0
StepHypRef Expression
1 elfir 7164 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  F ) )
2 filfi 17548 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( fi `  F )  =  F )
32adantr 453 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  -> 
( fi `  F
)  =  F )
41, 3eleqtrd 2360 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  F )
5 fileln0 17539 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  |^| A  e.  F )  ->  |^| A  =/=  (/) )
64, 5syldan 458 1  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447    C_ wss 3153   (/)c0 3456   |^|cint 3863   ` cfv 5221   Fincfn 6858   ficfi 7159   Filcfil 17534
This theorem is referenced by:  alexsublem  17732
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-en 6859  df-fin 6862  df-fi 7160  df-fbas 17514  df-fil 17535
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