Theorem elcls 23187

Description: Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
elcls	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑃   𝑥,𝑆   𝑥,𝑋

Proof of Theorem elcls

Step	Hyp	Ref	Expression
1		clscld.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
2	1	cmclsopn 23176	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
3	2	3adant3 1148	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
4	3	adantr 485	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5		eldif 3917	. . . . . . 7 ⊢ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ↔ (𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
6	5	biimpri 231	. . . . . 6 ⊢ ((𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
7	6	3ad2antl3 1204	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8		simpr 489	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
9	1	sscls 23170	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
10	8, 9	ssind 4195	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11		dfin4 4233	. . . . . . . . . 10 ⊢ (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
12	10, 11	sseqtrdi 3979	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13		reldisj 4410	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
14	13	adantl 486	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
15	12, 14	mpbird 260	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
16		nne 2964	. . . . . . . . 9 ⊢ (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17		incom 4164	. . . . . . . . . 10 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
18	17	eqeq1i 2770	. . . . . . . . 9 ⊢ (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
19	16, 18	bitri 278	. . . . . . . 8 ⊢ (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
20	15, 19	sylibr 237	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
21	20	3adant3 1148	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
22	21	adantr 485	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
23		eleq2 2854	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24		ineq1 4168	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
25	24	neeq1d 3019	. . . . . . . 8 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑥 ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
26	25	notbid 321	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (¬ (𝑥 ∩ 𝑆) ≠ ∅ ↔ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
27	23, 26	anbi12d 643	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
28	27	rspcev 3584	. . . . 5 ⊢ (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅))
29	4, 7, 22, 28	syl12anc 849	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅))
30		incom 4164	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ 𝑥) = (𝑥 ∩ 𝑆)
31	30	eqeq1i 2770	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑆) = ∅)
32		df-ne 2961	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝑆) ≠ ∅ ↔ ¬ (𝑥 ∩ 𝑆) = ∅)
33	32	con2bii 360	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑆) = ∅ ↔ ¬ (𝑥 ∩ 𝑆) ≠ ∅)
34	31, 33	bitri 278	. . . . . . . . . . 11 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ ¬ (𝑥 ∩ 𝑆) ≠ ∅)
35	1	opncld 23147	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
36	35	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
37		reldisj 4410	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → ((𝑆 ∩ 𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
38	37	biimpa 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝑋 ∧ (𝑆 ∩ 𝑥) = ∅) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
39	38	ad4ant24 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
40	1	clsss2 23186	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋 ∖ 𝑥))
41	36, 39, 40	syl2an2r 697	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋 ∖ 𝑥))
42	41	sseld 3938	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋 ∖ 𝑥)))
43		eldifn 4088	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (𝑋 ∖ 𝑥) → ¬ 𝑃 ∈ 𝑥)
44	42, 43	syl6 36	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ¬ 𝑃 ∈ 𝑥))
45	44	con2d 135	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ 𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
46	34, 45	sylan2br 606	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → (𝑃 ∈ 𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
47	46	exp31 424	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝐽 → (¬ (𝑥 ∩ 𝑆) ≠ ∅ → (𝑃 ∈ 𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
48	47	com34 92	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝐽 → (𝑃 ∈ 𝑥 → (¬ (𝑥 ∩ 𝑆) ≠ ∅ → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
49	48	imp4a 427	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝐽 → ((𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
50	49	rexlimdv 3164	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
51	50	imp 411	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
52	51	3adantl3 1185	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
53	29, 52	impbida 812	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)))
54		rexanali 3119	. . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) ↔ ¬ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅))
55	53, 54	bitrdi 290	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ¬ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))
56	55	con4bid 320	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089   ∖ cdif 3904   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  ∪ cuni 4867  ‘cfv 6525  Topctop 23007  Clsdccld 23130  clsccl 23132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-top 23008  df-cld 23133  df-ntr 23134  df-cls 23135

This theorem is referenced by:  elcls2  23188  clsndisj  23189  elcls3  23197  neindisj2  23237  islp3  23260  lmcls  23416  1stccnp  23576  txcls  23718  dfac14lem  23731  fclsopn  24128  metdseq0  24969  qndenserrn  46872