Theorem fclsneii 23961

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 23955	. . . . 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 2736	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	neii1 23050	. . . . 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 23001	. . . 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 4196	. 2 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (((int‘𝐽)‘𝑁) ∩ 𝑆) ⊆ (𝑁 ∩ 𝑆))
11	5	ntropn 22993	. . . 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 23960	. . . . . . . 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 4765	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → {𝐴} ⊆ ∪ 𝐽)
16	5	neiint 23048	. . . . . 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 4740	. . . . 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 23959	. . 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 4356	. 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 2113   ≠ wne 2932   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  {csn 4580  ∪ cuni 4863  ‘cfv 6492  (class class class)co 7358  Topctop 22837  intcnt 22961  neicnei 23041   fClus cfcls 23880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-fbas 21306  df-top 22838  df-topon 22855  df-cld 22963  df-ntr 22964  df-cls 22965  df-nei 23042  df-fil 23790  df-fcls 23885

This theorem is referenced by:  fclsnei  23963  fclsfnflim  23971