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Theorem hausflimlem 18011
Description: If  A and  B are both limits of the same filter, then all neighborhoods of  A and  B intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)
Assertion
Ref Expression
hausflimlem  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  ( U  i^i  V )  =/=  (/) )

Proof of Theorem hausflimlem
StepHypRef Expression
1 simp1l 981 . . 3  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  A  e.  ( J  fLim  F ) )
2 eqid 2436 . . . 4  |-  U. J  =  U. J
32flimfil 18001 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
41, 3syl 16 . 2  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  F  e.  ( Fil `  U. J
) )
5 flimtop 17997 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
61, 5syl 16 . . . 4  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  J  e.  Top )
7 simp2l 983 . . . 4  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  U  e.  J )
8 simp3l 985 . . . 4  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  A  e.  U )
9 opnneip 17183 . . . 4  |-  ( ( J  e.  Top  /\  U  e.  J  /\  A  e.  U )  ->  U  e.  ( ( nei `  J ) `
 { A }
) )
106, 7, 8, 9syl3anc 1184 . . 3  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  U  e.  ( ( nei `  J
) `  { A } ) )
11 flimnei 17999 . . 3  |-  ( ( A  e.  ( J 
fLim  F )  /\  U  e.  ( ( nei `  J
) `  { A } ) )  ->  U  e.  F )
121, 10, 11syl2anc 643 . 2  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  U  e.  F )
13 simp1r 982 . . 3  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  B  e.  ( J  fLim  F ) )
14 simp2r 984 . . . 4  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  V  e.  J )
15 simp3r 986 . . . 4  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  B  e.  V )
16 opnneip 17183 . . . 4  |-  ( ( J  e.  Top  /\  V  e.  J  /\  B  e.  V )  ->  V  e.  ( ( nei `  J ) `
 { B }
) )
176, 14, 15, 16syl3anc 1184 . . 3  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  V  e.  ( ( nei `  J
) `  { B } ) )
18 flimnei 17999 . . 3  |-  ( ( B  e.  ( J 
fLim  F )  /\  V  e.  ( ( nei `  J
) `  { B } ) )  ->  V  e.  F )
1913, 17, 18syl2anc 643 . 2  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  V  e.  F )
20 filinn0 17892 . 2  |-  ( ( F  e.  ( Fil `  U. J )  /\  U  e.  F  /\  V  e.  F )  ->  ( U  i^i  V
)  =/=  (/) )
214, 12, 19, 20syl3anc 1184 1  |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
)  ->  ( U  i^i  V )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2599    i^i cin 3319   (/)c0 3628   {csn 3814   U.cuni 4015   ` cfv 5454  (class class class)co 6081   Topctop 16958   neicnei 17161   Filcfil 17877    fLim cflim 17966
This theorem is referenced by:  hausflimi  18012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-fbas 16699  df-top 16963  df-nei 17162  df-fil 17878  df-flim 17971
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