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Theorem elcls 20629
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 20618 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
323adant3 1073 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
43adantr 479 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5 eldif 3549 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
65biimpri 216 . . . . . 6 ((𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
763ad2antl3 1217 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8 simpr 475 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆𝑋)
91sscls 20612 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
108, 9ssind 3798 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11 dfin4 3825 . . . . . . . . . 10 (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1210, 11syl6sseq 3613 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13 reldisj 3971 . . . . . . . . . 10 (𝑆𝑋 → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
1413adantl 480 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
1512, 14mpbird 245 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
16 nne 2785 . . . . . . . . 9 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17 incom 3766 . . . . . . . . . 10 ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1817eqeq1i 2614 . . . . . . . . 9 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
1916, 18bitri 262 . . . . . . . 8 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
2015, 19sylibr 222 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
21203adant3 1073 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
2221adantr 479 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
23 eleq2 2676 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃𝑥𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24 ineq1 3768 . . . . . . . . 9 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
2524neeq1d 2840 . . . . . . . 8 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑥𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
2625notbid 306 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (¬ (𝑥𝑆) ≠ ∅ ↔ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
2723, 26anbi12d 742 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
2827rspcev 3281 . . . . 5 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
294, 7, 22, 28syl12anc 1315 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
30 incom 3766 . . . . . . . . . . . . 13 (𝑆𝑥) = (𝑥𝑆)
3130eqeq1i 2614 . . . . . . . . . . . 12 ((𝑆𝑥) = ∅ ↔ (𝑥𝑆) = ∅)
32 df-ne 2781 . . . . . . . . . . . . 13 ((𝑥𝑆) ≠ ∅ ↔ ¬ (𝑥𝑆) = ∅)
3332con2bii 345 . . . . . . . . . . . 12 ((𝑥𝑆) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
3431, 33bitri 262 . . . . . . . . . . 11 ((𝑆𝑥) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
351opncld 20589 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
3635adantlr 746 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
3736adantr 479 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑋𝑥) ∈ (Clsd‘𝐽))
38 reldisj 3971 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋 → ((𝑆𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋𝑥)))
3938biimpa 499 . . . . . . . . . . . . . . . . 17 ((𝑆𝑋 ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
4039adantll 745 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
4140adantlr 746 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
421clsss2 20628 . . . . . . . . . . . . . . 15 (((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4337, 41, 42syl2anc 690 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4443sseld 3566 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋𝑥)))
45 eldifn 3694 . . . . . . . . . . . . 13 (𝑃 ∈ (𝑋𝑥) → ¬ 𝑃𝑥)
4644, 45syl6 34 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ¬ 𝑃𝑥))
4746con2d 127 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
4834, 47sylan2br 491 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ ¬ (𝑥𝑆) ≠ ∅) → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
4948exp31 627 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → (¬ (𝑥𝑆) ≠ ∅ → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
5049com34 88 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → (𝑃𝑥 → (¬ (𝑥𝑆) ≠ ∅ → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
5150imp4a 611 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
5251rexlimdv 3011 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
5352imp 443 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
54533adantl3 1211 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
5529, 54impbida 872 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)))
56 rexanali 2980 . . 3 (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ ¬ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅))
5755, 56syl6bb 274 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ¬ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
5857con4bid 305 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  cdif 3536  cin 3538  wss 3539  c0 3873   cuni 4366  cfv 5790  Topctop 20459  Clsdccld 20572  clsccl 20574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-top 20463  df-cld 20575  df-ntr 20576  df-cls 20577
This theorem is referenced by:  elcls2  20630  clsndisj  20631  elcls3  20639  neindisj2  20679  islp3  20702  lmcls  20858  1stccnp  21017  txcls  21159  dfac14lem  21172  fclsopn  21570  metdseq0  22396  qndenserrn  38999
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