Theorem elcls 22506

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 22495	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
3	2	3adant3 1132	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
4	3	adantr 481	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5		eldif 3954	. . . . . . 7 ⊢ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ↔ (𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
6	5	biimpri 227	. . . . . 6 ⊢ ((𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
7	6	3ad2antl3 1187	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8		simpr 485	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
9	1	sscls 22489	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
10	8, 9	ssind 4228	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11		dfin4 4263	. . . . . . . . . 10 ⊢ (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
12	10, 11	sseqtrdi 4028	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13		reldisj 4447	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
14	13	adantl 482	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
15	12, 14	mpbird 256	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
16		nne 2943	. . . . . . . . 9 ⊢ (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17		incom 4197	. . . . . . . . . 10 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
18	17	eqeq1i 2736	. . . . . . . . 9 ⊢ (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
19	16, 18	bitri 274	. . . . . . . 8 ⊢ (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
20	15, 19	sylibr 233	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
21	20	3adant3 1132	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
22	21	adantr 481	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
23		eleq2 2821	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24		ineq1 4201	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
25	24	neeq1d 2999	. . . . . . . 8 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑥 ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
26	25	notbid 317	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (¬ (𝑥 ∩ 𝑆) ≠ ∅ ↔ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
27	23, 26	anbi12d 631	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
28	27	rspcev 3609	. . . . 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 835	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅))
30		incom 4197	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ 𝑥) = (𝑥 ∩ 𝑆)
31	30	eqeq1i 2736	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑆) = ∅)
32		df-ne 2940	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝑆) ≠ ∅ ↔ ¬ (𝑥 ∩ 𝑆) = ∅)
33	32	con2bii 357	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑆) = ∅ ↔ ¬ (𝑥 ∩ 𝑆) ≠ ∅)
34	31, 33	bitri 274	. . . . . . . . . . 11 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ ¬ (𝑥 ∩ 𝑆) ≠ ∅)
35	1	opncld 22466	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
36	35	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
37		reldisj 4447	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → ((𝑆 ∩ 𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
38	37	biimpa 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝑋 ∧ (𝑆 ∩ 𝑥) = ∅) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
39	38	ad4ant24 752	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
40	1	clsss2 22505	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋 ∖ 𝑥))
41	36, 39, 40	syl2an2r 683	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋 ∖ 𝑥))
42	41	sseld 3977	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋 ∖ 𝑥)))
43		eldifn 4123	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (𝑋 ∖ 𝑥) → ¬ 𝑃 ∈ 𝑥)
44	42, 43	syl6 35	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ¬ 𝑃 ∈ 𝑥))
45	44	con2d 134	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ 𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
46	34, 45	sylan2br 595	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → (𝑃 ∈ 𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
47	46	exp31 420	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝐽 → (¬ (𝑥 ∩ 𝑆) ≠ ∅ → (𝑃 ∈ 𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
48	47	com34 91	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝐽 → (𝑃 ∈ 𝑥 → (¬ (𝑥 ∩ 𝑆) ≠ ∅ → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
49	48	imp4a 423	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝐽 → ((𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
50	49	rexlimdv 3152	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
51	50	imp 407	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
52	51	3adantl3 1168	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
53	29, 52	impbida 799	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)))
54		rexanali 3101	. . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) ↔ ¬ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅))
55	53, 54	bitrdi 286	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ¬ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))
56	55	con4bid 316	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∖ cdif 3941   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318  ∪ cuni 4901  ‘cfv 6532  Topctop 22324  Clsdccld 22449  clsccl 22451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-top 22325  df-cld 22452  df-ntr 22453  df-cls 22454

This theorem is referenced by:  elcls2  22507  clsndisj  22508  elcls3  22516  neindisj2  22556  islp3  22579  lmcls  22735  1stccnp  22895  txcls  23037  dfac14lem  23050  fclsopn  23447  metdseq0  24299  qndenserrn  44786