Theorem elcls 22989

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 22978	. . . . . . 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 480	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5		eldif 3908	. . . . . . 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 22972	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
10	8, 9	ssind 4190	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11		dfin4 4227	. . . . . . . . . 10 ⊢ (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
12	10, 11	sseqtrdi 3971	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13		reldisj 4402	. . . . . . . . . 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 2933	. . . . . . . . 9 ⊢ (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17		incom 4158	. . . . . . . . . 10 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
18	17	eqeq1i 2738	. . . . . . . . 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 1132	. . . . . 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 2822	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24		ineq1 4162	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
25	24	neeq1d 2988	. . . . . . . 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 3573	. . . . 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 836	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅))
30		incom 4158	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ 𝑥) = (𝑥 ∩ 𝑆)
31	30	eqeq1i 2738	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑆) = ∅)
32		df-ne 2930	. . . . . . . . . . . . 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 22949	. . . . . . . . . . . . . . . 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 4402	. . . . . . . . . . . . . . . . 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 22988	. . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) ∧ (𝑆 ∩ 𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋 ∖ 𝑥)))
43		eldifn 4081	. . . . . . . . . . . . 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 3132	. . . . . 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 800	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ¬ (𝑥 ∩ 𝑆) ≠ ∅)))
54		rexanali 3087	. . 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 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  ∪ cuni 4858  ‘cfv 6486  Topctop 22809  Clsdccld 22932  clsccl 22934

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-top 22810  df-cld 22935  df-ntr 22936  df-cls 22937

This theorem is referenced by:  elcls2  22990  clsndisj  22991  elcls3  22999  neindisj2  23039  islp3  23062  lmcls  23218  1stccnp  23378  txcls  23520  dfac14lem  23533  fclsopn  23930  metdseq0  24771  qndenserrn  46421