Theorem fclsneii 23911

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 1136	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐴 ∈ (𝐽 fClus 𝐹))
2		fclstop 23905	. . . . 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 1137	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑁 ∈ ((nei‘𝐽)‘{𝐴}))
5		eqid 2730	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	neii1 23000	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑁 ⊆ ∪ 𝐽)
7	3, 4, 6	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑁 ⊆ ∪ 𝐽)
8	5	ntrss2 22951	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
9	3, 7, 8	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
10	9	ssrind 4210	. 2 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆))
11	5	ntropn 22943	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
12	3, 7, 11	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
13	5	fclselbas 23910	. . . . . . . 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 4776	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → {𝐴} ⊆ ∪ 𝐽)
16	5	neiint 22998	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
17	3, 15, 7, 16	syl3anc 1373	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
18	4, 17	mpbid 232	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → {𝐴} ⊆ ((int‘𝐽)‘𝑁))
19		snssg 4750	. . . . 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 257	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐴 ∈ ((int‘𝐽)‘𝑁))
22		simp3 1138	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑆 ∈ 𝐹)
23		fclsopni 23909	. . 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 1374	. 2 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
25		ssn0 4370	. 2 ⊢ (((((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆) ∧ (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅) → (𝑁 ∩ 𝑆) ≠ ∅)
26	10, 24, 25	syl2anc 584	1 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∩ 𝑆) ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   ∈ wcel 2109   ≠ wne 2926   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  {csn 4592  ∪ cuni 4874  ‘cfv 6514  (class class class)co 7390  Topctop 22787  intcnt 22911  neicnei 22991   fClus cfcls 23830

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-fbas 21268  df-top 22788  df-topon 22805  df-cld 22913  df-ntr 22914  df-cls 22915  df-nei 22992  df-fil 23740  df-fcls 23835

This theorem is referenced by:  fclsnei  23913  fclsfnflim  23921