Theorem elcls 23081

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 23070	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
3	2	3adant3 1133	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
4	3	adantr 480	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5		eldif 3961	. . . . . . 7 ⊢ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ↔ (𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
6	5	biimpri 228	. . . . . 6 ⊢ ((𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
7	6	3ad2antl3 1188	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
9	1	sscls 23064	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
10	8, 9	ssind 4241	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11		dfin4 4278	. . . . . . . . . 10 ⊢ (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
12	10, 11	sseqtrdi 4024	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13		reldisj 4453	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
14	13	adantl 481	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
15	12, 14	mpbird 257	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
16		nne 2944	. . . . . . . . 9 ⊢ (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17		incom 4209	. . . . . . . . . 10 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
18	17	eqeq1i 2742	. . . . . . . . 9 ⊢ (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
19	16, 18	bitri 275	. . . . . . . 8 ⊢ (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
20	15, 19	sylibr 234	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
21	20	3adant3 1133	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
22	21	adantr 480	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
23		eleq2 2830	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24		ineq1 4213	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
25	24	neeq1d 3000	. . . . . . . 8 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑥 ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
26	25	notbid 318	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (¬ (𝑥 ∩ 𝑆) ≠ ∅ ↔ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
27	23, 26	anbi12d 632	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
28	27	rspcev 3622	. . . . 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 837	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅))
30		incom 4209	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ 𝑥) = (𝑥 ∩ 𝑆)
31	30	eqeq1i 2742	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑆) = ∅)
32		df-ne 2941	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝑆) ≠ ∅ ↔ ¬ (𝑥 ∩ 𝑆) = ∅)
33	32	con2bii 357	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑆) = ∅ ↔ ¬ (𝑥 ∩ 𝑆) ≠ ∅)
34	31, 33	bitri 275	. . . . . . . . . . 11 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ ¬ (𝑥 ∩ 𝑆) ≠ ∅)
35	1	opncld 23041	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
36	35	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
37		reldisj 4453	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → ((𝑆 ∩ 𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
38	37	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝑋 ∧ (𝑆 ∩ 𝑥) = ∅) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
39	38	ad4ant24 754	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
40	1	clsss2 23080	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋 ∖ 𝑥))
41	36, 39, 40	syl2an2r 685	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋 ∖ 𝑥))
42	41	sseld 3982	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋 ∖ 𝑥)))
43		eldifn 4132	. . . . . . . . . . . . 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 419	. . . . . . . . 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 422	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝐽 → ((𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
50	49	rexlimdv 3153	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
51	50	imp 406	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
52	51	3adantl3 1169	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
53	29, 52	impbida 801	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)))
54		rexanali 3102	. . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) ↔ ¬ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅))
55	53, 54	bitrdi 287	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ¬ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))
56	55	con4bid 317	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  Clsdccld 23024  clsccl 23026

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029

This theorem is referenced by:  elcls2  23082  clsndisj  23083  elcls3  23091  neindisj2  23131  islp3  23154  lmcls  23310  1stccnp  23470  txcls  23612  dfac14lem  23625  fclsopn  24022  metdseq0  24876  qndenserrn  46314