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Theorem filintn0 17572
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 7185 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  F ) )
2 filfi 17570 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( fi `  F )  =  F )
32adantr 451 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  -> 
( fi `  F
)  =  F )
41, 3eleqtrd 2372 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  F )
5 fileln0 17561 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  |^| A  e.  F )  ->  |^| A  =/=  (/) )
64, 5syldan 456 1  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   |^|cint 3878   ` cfv 5271   Fincfn 6879   ficfi 7180   Filcfil 17556
This theorem is referenced by:  alexsublem  17754
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-nel 2462  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-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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-fbas 17536  df-fil 17557
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