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Theorem hausflimlem 17964
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 2404 . . . 4  |-  U. J  =  U. J
32flimfil 17954 . . 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 17950 . . . . 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 17138 . . . 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 17952 . . 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 17138 . . . 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 17952 . . 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 17845 . 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 1721    =/= wne 2567    i^i cin 3279   (/)c0 3588   {csn 3774   U.cuni 3975   ` cfv 5413  (class class class)co 6040   Topctop 16913   neicnei 17116   Filcfil 17830    fLim cflim 17919
This theorem is referenced by:  hausflimi  17965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-fbas 16654  df-top 16918  df-nei 17117  df-fil 17831  df-flim 17924
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