Theorem fclsneii 21821

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 1061	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐴 ∈ (𝐽 fClus 𝐹))
2		fclstop 21815	. . . . 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 1062	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑁 ∈ ((nei‘𝐽)‘{𝐴}))
5		eqid 2622	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	neii1 20910	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑁 ⊆ ∪ 𝐽)
7	3, 4, 6	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑁 ⊆ ∪ 𝐽)
8	5	ntrss2 20861	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
9	3, 7, 8	syl2anc 693	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
10		ssrin 3838	. . 3 ⊢ (((int‘𝐽)‘𝑁) ⊆ 𝑁 → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆))
11	9, 10	syl 17	. 2 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆))
12	5	ntropn 20853	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
13	3, 7, 12	syl2anc 693	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
14	5	fclselbas 21820	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ ∪ 𝐽)
15	1, 14	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐴 ∈ ∪ 𝐽)
16	15	snssd 4340	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → {𝐴} ⊆ ∪ 𝐽)
17	5	neiint 20908	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
18	3, 16, 7, 17	syl3anc 1326	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
19	4, 18	mpbid 222	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → {𝐴} ⊆ ((int‘𝐽)‘𝑁))
20		snssg 4327	. . . . 5 ⊢ (𝐴 ∈ ∪ 𝐽 → (𝐴 ∈ ((int‘𝐽)‘𝑁) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
21	15, 20	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝐴 ∈ ((int‘𝐽)‘𝑁) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑁)))
22	19, 21	mpbird 247	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝐴 ∈ ((int‘𝐽)‘𝑁))
23		simp3 1063	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → 𝑆 ∈ 𝐹)
24		fclsopni 21819	. . 3 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (((int‘𝐽)‘𝑁) ∈ 𝐽 ∧ 𝐴 ∈ ((int‘𝐽)‘𝑁) ∧ 𝑆 ∈ 𝐹)) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
25	1, 13, 22, 23, 24	syl13anc 1328	. 2 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅)
26		ssn0 3976	. 2 ⊢ (((((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆) ∧ (((int‘𝐽)‘𝑁) ∩ 𝑆) ≠ ∅) → (𝑁 ∩ 𝑆) ≠ ∅)
27	11, 25, 26	syl2anc 693	1 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∩ 𝑆) ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   ∈ wcel 1990   ≠ wne 2794   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ cuni 4436  ‘cfv 5888  (class class class)co 6650  Topctop 20698  intcnt 20821  neicnei 20901   fClus cfcls 21740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-fil 21650  df-fcls 21745

This theorem is referenced by:  fclsnei  21823  fclsfnflim  21831