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Theorem hausflimlem 20992
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 1029 . . 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 2422 . . . 4  |-  U. J  =  U. J
32flimfil 20982 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
41, 3syl 17 . 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 20978 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
61, 5syl 17 . . . 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 1031 . . . 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 1033 . . . 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 20133 . . . 4  |-  ( ( J  e.  Top  /\  U  e.  J  /\  A  e.  U )  ->  U  e.  ( ( nei `  J ) `
 { A }
) )
106, 7, 8, 9syl3anc 1264 . . 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 20980 . . 3  |-  ( ( A  e.  ( J 
fLim  F )  /\  U  e.  ( ( nei `  J
) `  { A } ) )  ->  U  e.  F )
121, 10, 11syl2anc 665 . 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 1030 . . 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 1032 . . . 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 1034 . . . 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 20133 . . . 4  |-  ( ( J  e.  Top  /\  V  e.  J  /\  B  e.  V )  ->  V  e.  ( ( nei `  J ) `
 { B }
) )
176, 14, 15, 16syl3anc 1264 . . 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 20980 . . 3  |-  ( ( B  e.  ( J 
fLim  F )  /\  V  e.  ( ( nei `  J
) `  { B } ) )  ->  V  e.  F )
1913, 17, 18syl2anc 665 . 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 20873 . 2  |-  ( ( F  e.  ( Fil `  U. J )  /\  U  e.  F  /\  V  e.  F )  ->  ( U  i^i  V
)  =/=  (/) )
214, 12, 19, 20syl3anc 1264 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1872    =/= wne 2614    i^i cin 3435   (/)c0 3761   {csn 3998   U.cuni 4219   ` cfv 5601  (class class class)co 6305   Topctop 19915   neicnei 20111   Filcfil 20858    fLim cflim 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-fbas 18966  df-top 19919  df-nei 20112  df-fil 20859  df-flim 20952
This theorem is referenced by:  hausflimi  20993
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