Theorem elcls 21678

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 21667	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
3	2	3adant3 1129	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
4	3	adantr 484	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5		eldif 3891	. . . . . . 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 1184	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8		simpr 488	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
9	1	sscls 21661	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
10	8, 9	ssind 4159	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11		dfin4 4194	. . . . . . . . . 10 ⊢ (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
12	10, 11	sseqtrdi 3965	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13		reldisj 4359	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
14	13	adantl 485	. . . . . . . . 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 2991	. . . . . . . . 9 ⊢ (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17		incom 4128	. . . . . . . . . 10 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
18	17	eqeq1i 2803	. . . . . . . . 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 1129	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
22	21	adantr 484	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
23		eleq2 2878	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24		ineq1 4131	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
25	24	neeq1d 3046	. . . . . . . 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 633	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
28	27	rspcev 3571	. . . . 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 4128	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ 𝑥) = (𝑥 ∩ 𝑆)
31	30	eqeq1i 2803	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑆) = ∅)
32		df-ne 2988	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝑆) ≠ ∅ ↔ ¬ (𝑥 ∩ 𝑆) = ∅)
33	32	con2bii 361	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑆) = ∅ ↔ ¬ (𝑥 ∩ 𝑆) ≠ ∅)
34	31, 33	bitri 278	. . . . . . . . . . 11 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ ¬ (𝑥 ∩ 𝑆) ≠ ∅)
35	1	opncld 21638	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
36	35	adantlr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
37		reldisj 4359	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → ((𝑆 ∩ 𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
38	37	biimpa 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝑋 ∧ (𝑆 ∩ 𝑥) = ∅) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
39	38	ad4ant24 753	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
40	1	clsss2 21677	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋 ∖ 𝑥))
41	36, 39, 40	syl2an2r 684	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋 ∖ 𝑥))
42	41	sseld 3914	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋 ∖ 𝑥)))
43		eldifn 4055	. . . . . . . . . . . . 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 136	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ 𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
46	34, 45	sylan2br 597	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → (𝑃 ∈ 𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
47	46	exp31 423	. . . . . . . . 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 426	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝐽 → ((𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
50	49	rexlimdv 3242	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
51	50	imp 410	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
52	51	3adantl3 1165	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
53	29, 52	impbida 800	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)))
54		rexanali 3224	. . 3 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅) ↔ ¬ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅))
55	53, 54	syl6bb 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ∪ cuni 4800  ‘cfv 6324  Topctop 21498  Clsdccld 21621  clsccl 21623

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-top 21499  df-cld 21624  df-ntr 21625  df-cls 21626

This theorem is referenced by:  elcls2  21679  clsndisj  21680  elcls3  21688  neindisj2  21728  islp3  21751  lmcls  21907  1stccnp  22067  txcls  22209  dfac14lem  22222  fclsopn  22619  metdseq0  23459  qndenserrn  42941