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Theorem hausflimlem 17726
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 979 . . 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 2316 . . . 4  |-  U. J  =  U. J
32flimfil 17716 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
41, 3syl 15 . 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 17712 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
61, 5syl 15 . . . 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 981 . . . 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 983 . . . 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 16912 . . . 4  |-  ( ( J  e.  Top  /\  U  e.  J  /\  A  e.  U )  ->  U  e.  ( ( nei `  J ) `
 { A }
) )
106, 7, 8, 9syl3anc 1182 . . 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 17714 . . 3  |-  ( ( A  e.  ( J 
fLim  F )  /\  U  e.  ( ( nei `  J
) `  { A } ) )  ->  U  e.  F )
121, 10, 11syl2anc 642 . 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 980 . . 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 982 . . . 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 984 . . . 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 16912 . . . 4  |-  ( ( J  e.  Top  /\  V  e.  J  /\  B  e.  V )  ->  V  e.  ( ( nei `  J ) `
 { B }
) )
176, 14, 15, 16syl3anc 1182 . . 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 17714 . . 3  |-  ( ( B  e.  ( J 
fLim  F )  /\  V  e.  ( ( nei `  J
) `  { B } ) )  ->  V  e.  F )
1913, 17, 18syl2anc 642 . 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 17607 . 2  |-  ( ( F  e.  ( Fil `  U. J )  /\  U  e.  F  /\  V  e.  F )  ->  ( U  i^i  V
)  =/=  (/) )
214, 12, 19, 20syl3anc 1182 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 358    /\ w3a 934    e. wcel 1701    =/= wne 2479    i^i cin 3185   (/)c0 3489   {csn 3674   U.cuni 3864   ` cfv 5292  (class class class)co 5900   Topctop 16687   neicnei 16890   Filcfil 17592    fLim cflim 17681
This theorem is referenced by:  hausflimi  17727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-fbas 16429  df-top 16692  df-nei 16891  df-fil 17593  df-flim 17686
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