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Theorem hausflimlem 23987
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 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → (𝑈𝑉) ≠ ∅)

Proof of Theorem hausflimlem
StepHypRef Expression
1 simp1l 1198 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐴 ∈ (𝐽 fLim 𝐹))
2 eqid 2737 . . . 4 𝐽 = 𝐽
32flimfil 23977 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
41, 3syl 17 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐹 ∈ (Fil‘ 𝐽))
5 flimtop 23973 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
61, 5syl 17 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐽 ∈ Top)
7 simp2l 1200 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑈𝐽)
8 simp3l 1202 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐴𝑈)
9 opnneip 23127 . . . 4 ((𝐽 ∈ Top ∧ 𝑈𝐽𝐴𝑈) → 𝑈 ∈ ((nei‘𝐽)‘{𝐴}))
106, 7, 8, 9syl3anc 1373 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑈 ∈ ((nei‘𝐽)‘{𝐴}))
11 flimnei 23975 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑈 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑈𝐹)
121, 10, 11syl2anc 584 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑈𝐹)
13 simp1r 1199 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐵 ∈ (𝐽 fLim 𝐹))
14 simp2r 1201 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑉𝐽)
15 simp3r 1203 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐵𝑉)
16 opnneip 23127 . . . 4 ((𝐽 ∈ Top ∧ 𝑉𝐽𝐵𝑉) → 𝑉 ∈ ((nei‘𝐽)‘{𝐵}))
176, 14, 15, 16syl3anc 1373 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑉 ∈ ((nei‘𝐽)‘{𝐵}))
18 flimnei 23975 . . 3 ((𝐵 ∈ (𝐽 fLim 𝐹) ∧ 𝑉 ∈ ((nei‘𝐽)‘{𝐵})) → 𝑉𝐹)
1913, 17, 18syl2anc 584 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑉𝐹)
20 filinn0 23868 . 2 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑈𝐹𝑉𝐹) → (𝑈𝑉) ≠ ∅)
214, 12, 19, 20syl3anc 1373 1 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → (𝑈𝑉) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wcel 2108  wne 2940  cin 3950  c0 4333  {csn 4626   cuni 4907  cfv 6561  (class class class)co 7431  Topctop 22899  neicnei 23105  Filcfil 23853   fLim cflim 23942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-top 22900  df-nei 23106  df-fil 23854  df-flim 23947
This theorem is referenced by:  hausflimi  23988
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