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Theorem hausflimlem 21693
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 1083 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐴 ∈ (𝐽 fLim 𝐹))
2 eqid 2621 . . . 4 𝐽 = 𝐽
32flimfil 21683 . . 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 21679 . . . . 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 1085 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑈𝐽)
8 simp3l 1087 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐴𝑈)
9 opnneip 20833 . . . 4 ((𝐽 ∈ Top ∧ 𝑈𝐽𝐴𝑈) → 𝑈 ∈ ((nei‘𝐽)‘{𝐴}))
106, 7, 8, 9syl3anc 1323 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑈 ∈ ((nei‘𝐽)‘{𝐴}))
11 flimnei 21681 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑈 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑈𝐹)
121, 10, 11syl2anc 692 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑈𝐹)
13 simp1r 1084 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐵 ∈ (𝐽 fLim 𝐹))
14 simp2r 1086 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑉𝐽)
15 simp3r 1088 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝐵𝑉)
16 opnneip 20833 . . . 4 ((𝐽 ∈ Top ∧ 𝑉𝐽𝐵𝑉) → 𝑉 ∈ ((nei‘𝐽)‘{𝐵}))
176, 14, 15, 16syl3anc 1323 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑉 ∈ ((nei‘𝐽)‘{𝐵}))
18 flimnei 21681 . . 3 ((𝐵 ∈ (𝐽 fLim 𝐹) ∧ 𝑉 ∈ ((nei‘𝐽)‘{𝐵})) → 𝑉𝐹)
1913, 17, 18syl2anc 692 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → 𝑉𝐹)
20 filinn0 21574 . 2 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑈𝐹𝑉𝐹) → (𝑈𝑉) ≠ ∅)
214, 12, 19, 20syl3anc 1323 1 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → (𝑈𝑉) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1987  wne 2790  cin 3554  c0 3891  {csn 4148   cuni 4402  cfv 5847  (class class class)co 6604  Topctop 20617  neicnei 20811  Filcfil 21559   fLim cflim 21648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-fbas 19662  df-top 20621  df-nei 20812  df-fil 21560  df-flim 21653
This theorem is referenced by:  hausflimi  21694
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