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Theorem supnfcls 22322
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 4298 . 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 1172 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝑈𝐽)
4 simpl3 1173 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐴𝑈)
5 sseq2 3879 . . . . . 6 (𝑥 = (𝑋𝑈) → ((𝑋𝑈) ⊆ 𝑥 ↔ (𝑋𝑈) ⊆ (𝑋𝑈)))
6 difss 3994 . . . . . . 7 (𝑋𝑈) ⊆ 𝑋
7 simpl1 1171 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐽 ∈ (TopOn‘𝑋))
8 toponmax 21228 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
9 elpw2g 5097 . . . . . . . 8 (𝑋𝐽 → ((𝑋𝑈) ∈ 𝒫 𝑋 ↔ (𝑋𝑈) ⊆ 𝑋))
107, 8, 93syl 18 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → ((𝑋𝑈) ∈ 𝒫 𝑋 ↔ (𝑋𝑈) ⊆ 𝑋))
116, 10mpbiri 250 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ∈ 𝒫 𝑋)
12 ssidd 3876 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ⊆ (𝑋𝑈))
135, 11, 12elrabd 3592 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})
14 fclsopni 22317 . . . . 5 ((𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}) ∧ (𝑈𝐽𝐴𝑈 ∧ (𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅)
152, 3, 4, 13, 14syl13anc 1352 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅)
1615ex 405 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → (𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅))
1716necon2bd 2977 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → ((𝑈 ∩ (𝑋𝑈)) = ∅ → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})))
181, 17mpi 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 198  wa 387  w3a 1068   = wceq 1507  wcel 2048  wne 2961  {crab 3086  cdif 3822  cin 3824  wss 3825  c0 4173  𝒫 cpw 4416  cfv 6182  (class class class)co 6970  TopOnctopon 21212   fClus cfcls 22238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-fbas 20234  df-top 21196  df-topon 21213  df-cld 21321  df-ntr 21322  df-cls 21323  df-fil 22148  df-fcls 22243
This theorem is referenced by:  fclscf  22327
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