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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 ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑃   𝑥,𝑆   𝑥,𝑋

Proof of Theorem elcls
StepHypRef Expression
1 clscld.1 . . . . . . . 8 𝑋 = 𝐽
21cmclsopn 23070 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
323adant3 1133 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
43adantr 480 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5 eldif 3961 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
65biimpri 228 . . . . . 6 ((𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
763ad2antl3 1188 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8 simpr 484 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆𝑋)
91sscls 23064 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
108, 9ssind 4241 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11 dfin4 4278 . . . . . . . . . 10 (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1210, 11sseqtrdi 4024 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13 reldisj 4453 . . . . . . . . . 10 (𝑆𝑋 → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
1413adantl 481 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
1512, 14mpbird 257 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (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)))
1817eqeq1i 2742 . . . . . . . . 9 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
1916, 18bitri 275 . . . . . . . 8 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
2015, 19sylibr 234 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
21203adant3 1133 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
2221adantr 480 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
23 eleq2 2830 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃𝑥𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24 ineq1 4213 . . . . . . . . 9 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
2524neeq1d 3000 . . . . . . . 8 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑥𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
2625notbid 318 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (¬ (𝑥𝑆) ≠ ∅ ↔ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
2723, 26anbi12d 632 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
2827rspcev 3622 . . . . 5 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
294, 7, 22, 28syl12anc 837 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
30 incom 4209 . . . . . . . . . . . . 13 (𝑆𝑥) = (𝑥𝑆)
3130eqeq1i 2742 . . . . . . . . . . . 12 ((𝑆𝑥) = ∅ ↔ (𝑥𝑆) = ∅)
32 df-ne 2941 . . . . . . . . . . . . 13 ((𝑥𝑆) ≠ ∅ ↔ ¬ (𝑥𝑆) = ∅)
3332con2bii 357 . . . . . . . . . . . 12 ((𝑥𝑆) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
3431, 33bitri 275 . . . . . . . . . . 11 ((𝑆𝑥) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
351opncld 23041 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
3635adantlr 715 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
37 reldisj 4453 . . . . . . . . . . . . . . . . 17 (𝑆𝑋 → ((𝑆𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋𝑥)))
3837biimpa 476 . . . . . . . . . . . . . . . 16 ((𝑆𝑋 ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
3938ad4ant24 754 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
401clsss2 23080 . . . . . . . . . . . . . . 15 (((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4136, 39, 40syl2an2r 685 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4241sseld 3982 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋𝑥)))
43 eldifn 4132 . . . . . . . . . . . . 13 (𝑃 ∈ (𝑋𝑥) → ¬ 𝑃𝑥)
4442, 43syl6 35 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ¬ 𝑃𝑥))
4544con2d 134 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
4634, 45sylan2br 595 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ ¬ (𝑥𝑆) ≠ ∅) → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
4746exp31 419 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → (¬ (𝑥𝑆) ≠ ∅ → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
4847com34 91 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → (𝑃𝑥 → (¬ (𝑥𝑆) ≠ ∅ → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
4948imp4a 422 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
5049rexlimdv 3153 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
5150imp 406 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
52513adantl3 1169 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
5329, 52impbida 801 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)))
54 rexanali 3102 . . 3 (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ ¬ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅))
5553, 54bitrdi 287 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ¬ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
5655con4bid 317 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
Colors of variables: wff setvar class
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
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