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Theorem elcls 20687
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 20676 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
323adant3 1074 . . . . . 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 3550 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
65biimpri 217 . . . . . 6 ((𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
763ad2antl3 1218 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8 simpr 476 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆𝑋)
91sscls 20670 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
108, 9ssind 3799 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11 dfin4 3826 . . . . . . . . . 10 (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1210, 11syl6sseq 3614 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13 reldisj 3972 . . . . . . . . . 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 246 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
16 nne 2786 . . . . . . . . 9 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17 incom 3767 . . . . . . . . . 10 ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1817eqeq1i 2615 . . . . . . . . 9 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
1916, 18bitri 263 . . . . . . . 8 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
2015, 19sylibr 223 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
21203adant3 1074 . . . . . 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 2677 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃𝑥𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24 ineq1 3769 . . . . . . . . 9 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
2524neeq1d 2841 . . . . . . . 8 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑥𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
2625notbid 307 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (¬ (𝑥𝑆) ≠ ∅ ↔ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
2723, 26anbi12d 743 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
2827rspcev 3282 . . . . 5 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
294, 7, 22, 28syl12anc 1316 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
30 incom 3767 . . . . . . . . . . . . 13 (𝑆𝑥) = (𝑥𝑆)
3130eqeq1i 2615 . . . . . . . . . . . 12 ((𝑆𝑥) = ∅ ↔ (𝑥𝑆) = ∅)
32 df-ne 2782 . . . . . . . . . . . . 13 ((𝑥𝑆) ≠ ∅ ↔ ¬ (𝑥𝑆) = ∅)
3332con2bii 346 . . . . . . . . . . . 12 ((𝑥𝑆) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
3431, 33bitri 263 . . . . . . . . . . 11 ((𝑆𝑥) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
351opncld 20647 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
3635adantlr 747 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
3736adantr 480 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑋𝑥) ∈ (Clsd‘𝐽))
38 reldisj 3972 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋 → ((𝑆𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋𝑥)))
3938biimpa 500 . . . . . . . . . . . . . . . . 17 ((𝑆𝑋 ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
4039adantll 746 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
4140adantlr 747 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
421clsss2 20686 . . . . . . . . . . . . . . 15 (((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4337, 41, 42syl2anc 691 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4443sseld 3567 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋𝑥)))
45 eldifn 3695 . . . . . . . . . . . . 13 (𝑃 ∈ (𝑋𝑥) → ¬ 𝑃𝑥)
4644, 45syl6 34 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ¬ 𝑃𝑥))
4746con2d 128 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
4834, 47sylan2br 492 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ ¬ (𝑥𝑆) ≠ ∅) → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
4948exp31 628 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → (¬ (𝑥𝑆) ≠ ∅ → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
5049com34 89 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → (𝑃𝑥 → (¬ (𝑥𝑆) ≠ ∅ → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
5150imp4a 612 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
5251rexlimdv 3012 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
5352imp 444 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
54533adantl3 1212 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
5529, 54impbida 873 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)))
56 rexanali 2981 . . 3 (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ ¬ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅))
5755, 56syl6bb 275 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ¬ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
5857con4bid 306 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  cdif 3537  cin 3539  wss 3540  c0 3874   cuni 4372  cfv 5804  Topctop 20517  Clsdccld 20630  clsccl 20632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-ntr 20634  df-cls 20635
This theorem is referenced by:  elcls2  20688  clsndisj  20689  elcls3  20697  neindisj2  20737  islp3  20760  lmcls  20916  1stccnp  21075  txcls  21217  dfac14lem  21230  fclsopn  21628  metdseq0  22465  qndenserrn  39195
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