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Theorem supnfcls 21747
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 4017 . 2 (𝑈 ∩ (𝑋𝑈)) = ∅
2 simpr 477 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}))
3 simpl2 1063 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝑈𝐽)
4 simpl3 1064 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐴𝑈)
5 difss 3720 . . . . . . 7 (𝑋𝑈) ⊆ 𝑋
6 simpl1 1062 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐽 ∈ (TopOn‘𝑋))
7 toponmax 20652 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
8 elpw2g 4792 . . . . . . . 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 3608 . . . . . . 7 (𝑋𝑈) ⊆ (𝑋𝑈)
1211a1i 11 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ⊆ (𝑋𝑈))
13 sseq2 3611 . . . . . . 7 (𝑥 = (𝑋𝑈) → ((𝑋𝑈) ⊆ 𝑥 ↔ (𝑋𝑈) ⊆ (𝑋𝑈)))
1413elrab 3350 . . . . . 6 ((𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥} ↔ ((𝑋𝑈) ∈ 𝒫 𝑋 ∧ (𝑋𝑈) ⊆ (𝑋𝑈)))
1510, 12, 14sylanbrc 697 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})
16 fclsopni 21742 . . . . 5 ((𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}) ∧ (𝑈𝐽𝐴𝑈 ∧ (𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅)
172, 3, 4, 15, 16syl13anc 1325 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅)
1817ex 450 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → (𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅))
1918necon2bd 2806 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  {crab 2911  cdif 3556  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  cfv 5852  (class class class)co 6610  TopOnctopon 20647   fClus cfcls 21663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-fbas 19675  df-top 20631  df-topon 20648  df-cld 20746  df-ntr 20747  df-cls 20748  df-fil 21573  df-fcls 21668
This theorem is referenced by:  fclscf  21752
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