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Theorem hausflimlem 17674
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 2283 . . . 4  |-  U. J  =  U. J
32flimfil 17664 . . 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 17660 . . . . 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 16856 . . . 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 17662 . . 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 16856 . . . 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 17662 . . 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 17555 . 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 1684    =/= wne 2446    i^i cin 3151   (/)c0 3455   {csn 3640   U.cuni 3827   ` cfv 5255  (class class class)co 5858   Topctop 16631   neicnei 16834   Filcfil 17540    fLim cflim 17629
This theorem is referenced by:  hausflimi  17675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-nei 16835  df-fbas 17520  df-fil 17541  df-flim 17634
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