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Theorem supnfcls 23169
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 4411 . 2 (𝑈 ∩ (𝑋𝑈)) = ∅
2 simpr 485 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}))
3 simpl2 1191 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝑈𝐽)
4 simpl3 1192 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐴𝑈)
5 sseq2 3952 . . . . . 6 (𝑥 = (𝑋𝑈) → ((𝑋𝑈) ⊆ 𝑥 ↔ (𝑋𝑈) ⊆ (𝑋𝑈)))
6 difss 4071 . . . . . . 7 (𝑋𝑈) ⊆ 𝑋
7 simpl1 1190 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → 𝐽 ∈ (TopOn‘𝑋))
8 toponmax 22073 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
9 elpw2g 5272 . . . . . . . 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 257 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ∈ 𝒫 𝑋)
12 ssidd 3949 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ⊆ (𝑋𝑈))
135, 11, 12elrabd 3628 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})
14 fclsopni 23164 . . . . 5 ((𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}) ∧ (𝑈𝐽𝐴𝑈 ∧ (𝑋𝑈) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅)
152, 3, 4, 13, 14syl13anc 1371 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) ∧ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥})) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅)
1615ex 413 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → (𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}) → (𝑈 ∩ (𝑋𝑈)) ≠ ∅))
1716necon2bd 2961 . 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 205  wa 396  w3a 1086   = wceq 1542  wcel 2110  wne 2945  {crab 3070  cdif 3889  cin 3891  wss 3892  c0 4262  𝒫 cpw 4539  cfv 6432  (class class class)co 7271  TopOnctopon 22057   fClus cfcls 23085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-fbas 20592  df-top 22041  df-topon 22058  df-cld 22168  df-ntr 22169  df-cls 22170  df-fil 22995  df-fcls 23090
This theorem is referenced by:  fclscf  23174
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