Theorem hausflimlem 23038

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)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑈 ∩ 𝑉) ≠ ∅)

Proof of Theorem hausflimlem

Step	Hyp	Ref	Expression
1		simp1l 1195	. . 3 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ (𝐽 fLim 𝐹))
2		eqid 2738	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimfil 23028	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
4	1, 3	syl 17	. 2 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐹 ∈ (Fil‘∪ 𝐽))
5		flimtop 23024	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
6	1, 5	syl 17	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐽 ∈ Top)
7		simp2l 1197	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝑈 ∈ 𝐽)
8		simp3l 1199	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑈)
9		opnneip 22178	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → 𝑈 ∈ ((nei‘𝐽)‘{𝐴}))
10	6, 7, 8, 9	syl3anc 1369	. . 3 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝑈 ∈ ((nei‘𝐽)‘{𝐴}))
11		flimnei 23026	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑈 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑈 ∈ 𝐹)
12	1, 10, 11	syl2anc 583	. 2 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝑈 ∈ 𝐹)
13		simp1r 1196	. . 3 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ (𝐽 fLim 𝐹))
14		simp2r 1198	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝑉 ∈ 𝐽)
15		simp3r 1200	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
16		opnneip 22178	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑉 ∈ 𝐽 ∧ 𝐵 ∈ 𝑉) → 𝑉 ∈ ((nei‘𝐽)‘{𝐵}))
17	6, 14, 15, 16	syl3anc 1369	. . 3 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝑉 ∈ ((nei‘𝐽)‘{𝐵}))
18		flimnei 23026	. . 3 ⊢ ((𝐵 ∈ (𝐽 fLim 𝐹) ∧ 𝑉 ∈ ((nei‘𝐽)‘{𝐵})) → 𝑉 ∈ 𝐹)
19	13, 17, 18	syl2anc 583	. 2 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝑉 ∈ 𝐹)
20		filinn0 22919	. 2 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑈 ∈ 𝐹 ∧ 𝑉 ∈ 𝐹) → (𝑈 ∩ 𝑉) ≠ ∅)
21	4, 12, 19, 20	syl3anc 1369	1 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑈 ∩ 𝑉) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2108   ≠ wne 2942   ∩ cin 3882  ∅c0 4253  {csn 4558  ∪ cuni 4836  ‘cfv 6418  (class class class)co 7255  Topctop 21950  neicnei 22156  Filcfil 22904   fLim cflim 22993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-fbas 20507  df-top 21951  df-nei 22157  df-fil 22905  df-flim 22998

This theorem is referenced by:  hausflimi  23039