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Theorem clsndisj 21601
Description: Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clsndisj (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑈𝐽𝑃𝑈)) → (𝑈𝑆) ≠ ∅)

Proof of Theorem clsndisj
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp1 1130 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
2 simp2 1131 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝑋)
3 clscld.1 . . . . . 6 𝑋 = 𝐽
43clsss3 21585 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
54sseld 3969 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃𝑋))
653impia 1111 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃𝑋)
7 simp3 1132 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ ((cls‘𝐽)‘𝑆))
83elcls 21599 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
98biimpa 477 . . 3 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅))
101, 2, 6, 7, 9syl31anc 1367 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅))
11 eleq2 2905 . . . . 5 (𝑥 = 𝑈 → (𝑃𝑥𝑃𝑈))
12 ineq1 4184 . . . . . 6 (𝑥 = 𝑈 → (𝑥𝑆) = (𝑈𝑆))
1312neeq1d 3079 . . . . 5 (𝑥 = 𝑈 → ((𝑥𝑆) ≠ ∅ ↔ (𝑈𝑆) ≠ ∅))
1411, 13imbi12d 346 . . . 4 (𝑥 = 𝑈 → ((𝑃𝑥 → (𝑥𝑆) ≠ ∅) ↔ (𝑃𝑈 → (𝑈𝑆) ≠ ∅)))
1514rspccv 3623 . . 3 (∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅) → (𝑈𝐽 → (𝑃𝑈 → (𝑈𝑆) ≠ ∅)))
1615imp32 419 . 2 ((∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅) ∧ (𝑈𝐽𝑃𝑈)) → (𝑈𝑆) ≠ ∅)
1710, 16sylan 580 1 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑈𝐽𝑃𝑈)) → (𝑈𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3020  wral 3142  cin 3938  wss 3939  c0 4294   cuni 4836  cfv 6351  Topctop 21419  clsccl 21544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-top 21420  df-cld 21545  df-ntr 21546  df-cls 21547
This theorem is referenced by:  neindisj  21643  clsconn  21956  txcls  22130  ptclsg  22141  flimsncls  22512  hauspwpwf1  22513  met2ndci  23049  metdseq0  23379  heibor1lem  34957
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