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Theorem supnfcls 21871
Description: The filter of supersets of 𝑋𝑈 does not cluster at any point of the open set 𝑈. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
supnfcls ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋   𝑥,𝑈
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem supnfcls
StepHypRef Expression
1 disjdif 4073 . 2 (𝑈 ∩ (𝑋𝑈)) = ∅
2 simpr 476 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}))
3 simpl2 1085 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝑈𝐽)
4 simpl3 1086 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐴𝑈)
5 difss 3770 . . . . . . 7 (𝑋𝑈) ⊆ 𝑋
6 simpl1 1084 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐽 ∈ (TopOn‘𝑋))
7 toponmax 20778 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
8 elpw2g 4857 . . . . . . . 8 (𝑋𝐽 → ((𝑋𝑈) ∈ 𝒫 𝑋 ↔ (𝑋𝑈) ⊆ 𝑋))
96, 7, 83syl 18 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → ((𝑋𝑈) ∈ 𝒫 𝑋 ↔ (𝑋𝑈) ⊆ 𝑋))
105, 9mpbiri 248 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ∈ 𝒫 𝑋)
11 ssid 3657 . . . . . . 7 (𝑋𝑈) ⊆ (𝑋𝑈)
1211a1i 11 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ⊆ (𝑋𝑈))
13 sseq2 3660 . . . . . . 7 (𝑥 = (𝑋𝑈) → ((𝑋𝑈) ⊆ 𝑥 ↔ (𝑋𝑈) ⊆ (𝑋𝑈)))
1413elrab 3396 . . . . . 6 ((𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥} ↔ ((𝑋𝑈) ∈ 𝒫 𝑋 ∧ (𝑋𝑈) ⊆ (𝑋𝑈)))
1510, 12, 14sylanbrc 699 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})
16 fclsopni 21866 . . . . 5 ((𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}) ∧ (𝑈𝐽𝐴𝑈 ∧ (𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅)
172, 3, 4, 15, 16syl13anc 1368 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅)
1817ex 449 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → (𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅))
1918necon2bd 2839 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → ((𝑈 ∩ (𝑋𝑈)) = ∅ → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})))
201, 19mpi 20 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  {crab 2945  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  cfv 5926  (class class class)co 6690  TopOnctopon 20763   fClus cfcls 21787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-fbas 19791  df-top 20747  df-topon 20764  df-cld 20871  df-ntr 20872  df-cls 20873  df-fil 21697  df-fcls 21792
This theorem is referenced by:  fclscf  21876
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