Theorem fclsneii 23076

Description: A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclsneii	⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∩ 𝑆) ≠ ∅)

Proof of Theorem fclsneii

Step	Hyp	Ref	Expression
1		simp1 1134	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐴 ∈ (𝐽 fClus 𝐹))
2		fclstop 23070	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
3	1, 2	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐽 ∈ Top)
4		simp2 1135	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑁 ∈ ((nei‘𝐽)‘{𝐴}))
5		eqid 2738	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	neii1 22165	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑁 ⊆ ∪ 𝐽)
7	3, 4, 6	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑁 ⊆ ∪ 𝐽)
8	5	ntrss2 22116	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
9	3, 7, 8	syl2anc 583	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
10	9	ssrind 4166	. 2 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆))
11	5	ntropn 22108	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
12	3, 7, 11	syl2anc 583	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
13	5	fclselbas 23075	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ ∪ 𝐽)
14	1, 13	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐴 ∈ ∪ 𝐽)
15	14	snssd 4739	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → {𝐴} ⊆ ∪ 𝐽)
16	5	neiint 22163	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
17	3, 15, 7, 16	syl3anc 1369	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
18	4, 17	mpbid 231	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → {𝐴} ⊆ ((int‘𝐽)‘𝑁))
19		snssg 4715	. . . . 5 ⊢ (𝐴 ∈ ∪ 𝐽 → (𝐴 ∈ ((int‘𝐽)‘𝑁) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
20	14, 19	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝐴 ∈ ((int‘𝐽)‘𝑁) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
21	18, 20	mpbird 256	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐴 ∈ ((int‘𝐽)‘𝑁))
22		simp3 1136	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑆 ∈ 𝐹)
23		fclsopni 23074	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (((int‘𝐽)‘𝑁) ∈ 𝐽 ∧ 𝐴 ∈ ((int‘𝐽)‘𝑁) ∧ 𝑆 ∈ 𝐹)) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
24	1, 12, 21, 22, 23	syl13anc 1370	. 2 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
25		ssn0 4331	. 2 ⊢ (((((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆) ∧ (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅) → (𝑁 ∩ 𝑆) ≠ ∅)
26	10, 24, 25	syl2anc 583	1 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∩ 𝑆) ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   ∈ wcel 2108   ≠ wne 2942   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ cuni 4836  ‘cfv 6418  (class class class)co 7255  Topctop 21950  intcnt 22076  neicnei 22156   fClus cfcls 22995

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  ax-un 7566

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-int 4877  df-iun 4923  df-iin 4924  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-topon 21968  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-fil 22905  df-fcls 23000

This theorem is referenced by:  fclsnei  23078  fclsfnflim  23086