Theorem fclsneii 23904

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 23898	. . . . 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 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	neii1 22993	. . . . 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 22944	. . . 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 4207	. 2 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆))
11	5	ntropn 22936	. . . 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 23903	. . . . . . . 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 4773	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → {𝐴} ⊆ ∪ 𝐽)
16	5	neiint 22991	. . . . . 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 4747	. . . . 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 23902	. . 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 4367	. 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 2925   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  {csn 4589  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387  Topctop 22780  intcnt 22904  neicnei 22984   fClus cfcls 23823

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-fbas 21261  df-top 22781  df-topon 22798  df-cld 22906  df-ntr 22907  df-cls 22908  df-nei 22985  df-fil 23733  df-fcls 23828

This theorem is referenced by:  fclsnei  23906  fclsfnflim  23914