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Theorem elcls 22224
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 22213 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
323adant3 1131 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
43adantr 481 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5 eldif 3897 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
65biimpri 227 . . . . . 6 ((𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
763ad2antl3 1186 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8 simpr 485 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆𝑋)
91sscls 22207 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
108, 9ssind 4166 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11 dfin4 4201 . . . . . . . . . 10 (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1210, 11sseqtrdi 3971 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13 reldisj 4385 . . . . . . . . . 10 (𝑆𝑋 → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
1413adantl 482 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
1512, 14mpbird 256 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
16 nne 2947 . . . . . . . . 9 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17 incom 4135 . . . . . . . . . 10 ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1817eqeq1i 2743 . . . . . . . . 9 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
1916, 18bitri 274 . . . . . . . 8 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
2015, 19sylibr 233 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
21203adant3 1131 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
2221adantr 481 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
23 eleq2 2827 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃𝑥𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24 ineq1 4139 . . . . . . . . 9 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
2524neeq1d 3003 . . . . . . . 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 631 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
2827rspcev 3561 . . . . 5 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
294, 7, 22, 28syl12anc 834 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
30 incom 4135 . . . . . . . . . . . . 13 (𝑆𝑥) = (𝑥𝑆)
3130eqeq1i 2743 . . . . . . . . . . . 12 ((𝑆𝑥) = ∅ ↔ (𝑥𝑆) = ∅)
32 df-ne 2944 . . . . . . . . . . . . 13 ((𝑥𝑆) ≠ ∅ ↔ ¬ (𝑥𝑆) = ∅)
3332con2bii 358 . . . . . . . . . . . 12 ((𝑥𝑆) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
3431, 33bitri 274 . . . . . . . . . . 11 ((𝑆𝑥) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
351opncld 22184 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
3635adantlr 712 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
37 reldisj 4385 . . . . . . . . . . . . . . . . 17 (𝑆𝑋 → ((𝑆𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋𝑥)))
3837biimpa 477 . . . . . . . . . . . . . . . 16 ((𝑆𝑋 ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
3938ad4ant24 751 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
401clsss2 22223 . . . . . . . . . . . . . . 15 (((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4136, 39, 40syl2an2r 682 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4241sseld 3920 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋𝑥)))
43 eldifn 4062 . . . . . . . . . . . . 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 420 . . . . . . . . 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 423 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
5049rexlimdv 3212 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
5150imp 407 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
52513adantl3 1167 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
5329, 52impbida 798 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)))
54 rexanali 3192 . . 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 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wral 3064  wrex 3065  cdif 3884  cin 3886  wss 3887  c0 4256   cuni 4839  cfv 6433  Topctop 22042  Clsdccld 22167  clsccl 22169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170  df-ntr 22171  df-cls 22172
This theorem is referenced by:  elcls2  22225  clsndisj  22226  elcls3  22234  neindisj2  22274  islp3  22297  lmcls  22453  1stccnp  22613  txcls  22755  dfac14lem  22768  fclsopn  23165  metdseq0  24017  qndenserrn  43840
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