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Theorem hausflimlem 24003
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 1196 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐴 ∈ (𝐽 fLim 𝐹))
2 eqid 2735 . . . 4 𝐽 = 𝐽
32flimfil 23993 . . 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 23989 . . . . 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 1198 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑈𝐽)
8 simp3l 1200 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐴𝑈)
9 opnneip 23143 . . . 4 ((𝐽 ∈ Top ∧ 𝑈𝐽𝐴𝑈) → 𝑈 ∈ ((nei‘𝐽)‘{𝐴}))
106, 7, 8, 9syl3anc 1370 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑈 ∈ ((nei‘𝐽)‘{𝐴}))
11 flimnei 23991 . . 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 1197 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐵 ∈ (𝐽 fLim 𝐹))
14 simp2r 1199 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑉𝐽)
15 simp3r 1201 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐵𝑉)
16 opnneip 23143 . . . 4 ((𝐽 ∈ Top ∧ 𝑉𝐽𝐵𝑉) → 𝑉 ∈ ((nei‘𝐽)‘{𝐵}))
176, 14, 15, 16syl3anc 1370 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑉 ∈ ((nei‘𝐽)‘{𝐵}))
18 flimnei 23991 . . 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 23884 . 2 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑈𝐹𝑉𝐹) → (𝑈𝑉) ≠ ∅)
214, 12, 19, 20syl3anc 1370 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 1086  wcel 2106  wne 2938  cin 3962  c0 4339  {csn 4631   cuni 4912  cfv 6563  (class class class)co 7431  Topctop 22915  neicnei 23121  Filcfil 23869   fLim cflim 23958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-top 22916  df-nei 23122  df-fil 23870  df-flim 23963
This theorem is referenced by:  hausflimi  24004
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