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Theorem dissnlocfin 21380
Description: The set of singletons is locally finite in the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.)
Hypothesis
Ref Expression
dissnref.c 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
Assertion
Ref Expression
dissnlocfin (𝑋𝑉𝐶 ∈ (LocFin‘𝒫 𝑋))
Distinct variable groups:   𝑢,𝐶,𝑥   𝑢,𝑉,𝑥   𝑢,𝑋,𝑥

Proof of Theorem dissnlocfin
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 20847 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2 eqidd 2652 . 2 (𝑋𝑉𝑋 = 𝑋)
3 snelpwi 4942 . . . . 5 (𝑧𝑋 → {𝑧} ∈ 𝒫 𝑋)
43adantl 481 . . . 4 ((𝑋𝑉𝑧𝑋) → {𝑧} ∈ 𝒫 𝑋)
5 vsnid 4242 . . . . 5 𝑧 ∈ {𝑧}
65a1i 11 . . . 4 ((𝑋𝑉𝑧𝑋) → 𝑧 ∈ {𝑧})
7 nfv 1883 . . . . . 6 𝑢(𝑋𝑉𝑧𝑋)
8 nfrab1 3152 . . . . . 6 𝑢{𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅}
9 nfcv 2793 . . . . . 6 𝑢{{𝑧}}
10 dissnref.c . . . . . . . . . 10 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
1110abeq2i 2764 . . . . . . . . 9 (𝑢𝐶 ↔ ∃𝑥𝑋 𝑢 = {𝑥})
1211anbi1i 731 . . . . . . . 8 ((𝑢𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ (∃𝑥𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
13 simpr 476 . . . . . . . . . . . . 13 (((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
14 simplr 807 . . . . . . . . . . . . . . . . . 18 ((((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥𝑧) → 𝑢 = {𝑥})
1514ineq1d 3846 . . . . . . . . . . . . . . . . 17 ((((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥𝑧) → (𝑢 ∩ {𝑧}) = ({𝑥} ∩ {𝑧}))
16 disjsn2 4279 . . . . . . . . . . . . . . . . . 18 (𝑥𝑧 → ({𝑥} ∩ {𝑧}) = ∅)
1716adantl 481 . . . . . . . . . . . . . . . . 17 ((((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥𝑧) → ({𝑥} ∩ {𝑧}) = ∅)
1815, 17eqtrd 2685 . . . . . . . . . . . . . . . 16 ((((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥𝑧) → (𝑢 ∩ {𝑧}) = ∅)
19 simp-4r 824 . . . . . . . . . . . . . . . . 17 ((((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥𝑧) → (𝑢 ∩ {𝑧}) ≠ ∅)
2019neneqd 2828 . . . . . . . . . . . . . . . 16 ((((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥𝑧) → ¬ (𝑢 ∩ {𝑧}) = ∅)
2118, 20pm2.65da 599 . . . . . . . . . . . . . . 15 (((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → ¬ 𝑥𝑧)
22 nne 2827 . . . . . . . . . . . . . . 15 𝑥𝑧𝑥 = 𝑧)
2321, 22sylib 208 . . . . . . . . . . . . . 14 (((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 = 𝑧)
2423sneqd 4222 . . . . . . . . . . . . 13 (((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} = {𝑧})
2513, 24eqtrd 2685 . . . . . . . . . . . 12 (((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑧})
2625r19.29an 3106 . . . . . . . . . . 11 ((((𝑋𝑉𝑧𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ ∃𝑥𝑋 𝑢 = {𝑥}) → 𝑢 = {𝑧})
2726an32s 863 . . . . . . . . . 10 ((((𝑋𝑉𝑧𝑋) ∧ ∃𝑥𝑋 𝑢 = {𝑥}) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) → 𝑢 = {𝑧})
2827anasss 680 . . . . . . . . 9 (((𝑋𝑉𝑧𝑋) ∧ (∃𝑥𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅)) → 𝑢 = {𝑧})
29 sneq 4220 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → {𝑥} = {𝑧})
3029eqeq2d 2661 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
3130rspcev 3340 . . . . . . . . . . 11 ((𝑧𝑋𝑢 = {𝑧}) → ∃𝑥𝑋 𝑢 = {𝑥})
3231adantll 750 . . . . . . . . . 10 (((𝑋𝑉𝑧𝑋) ∧ 𝑢 = {𝑧}) → ∃𝑥𝑋 𝑢 = {𝑥})
33 simpr 476 . . . . . . . . . . . . 13 (((𝑋𝑉𝑧𝑋) ∧ 𝑢 = {𝑧}) → 𝑢 = {𝑧})
3433ineq1d 3846 . . . . . . . . . . . 12 (((𝑋𝑉𝑧𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) = ({𝑧} ∩ {𝑧}))
35 inidm 3855 . . . . . . . . . . . 12 ({𝑧} ∩ {𝑧}) = {𝑧}
3634, 35syl6eq 2701 . . . . . . . . . . 11 (((𝑋𝑉𝑧𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) = {𝑧})
37 vex 3234 . . . . . . . . . . . . 13 𝑧 ∈ V
3837snnz 4340 . . . . . . . . . . . 12 {𝑧} ≠ ∅
3938a1i 11 . . . . . . . . . . 11 (((𝑋𝑉𝑧𝑋) ∧ 𝑢 = {𝑧}) → {𝑧} ≠ ∅)
4036, 39eqnetrd 2890 . . . . . . . . . 10 (((𝑋𝑉𝑧𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) ≠ ∅)
4132, 40jca 553 . . . . . . . . 9 (((𝑋𝑉𝑧𝑋) ∧ 𝑢 = {𝑧}) → (∃𝑥𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
4228, 41impbida 895 . . . . . . . 8 ((𝑋𝑉𝑧𝑋) → ((∃𝑥𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ 𝑢 = {𝑧}))
4312, 42syl5bb 272 . . . . . . 7 ((𝑋𝑉𝑧𝑋) → ((𝑢𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ 𝑢 = {𝑧}))
44 rabid 3145 . . . . . . 7 (𝑢 ∈ {𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ↔ (𝑢𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
45 velsn 4226 . . . . . . 7 (𝑢 ∈ {{𝑧}} ↔ 𝑢 = {𝑧})
4643, 44, 453bitr4g 303 . . . . . 6 ((𝑋𝑉𝑧𝑋) → (𝑢 ∈ {𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ↔ 𝑢 ∈ {{𝑧}}))
477, 8, 9, 46eqrd 3655 . . . . 5 ((𝑋𝑉𝑧𝑋) → {𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} = {{𝑧}})
48 snfi 8079 . . . . 5 {{𝑧}} ∈ Fin
4947, 48syl6eqel 2738 . . . 4 ((𝑋𝑉𝑧𝑋) → {𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)
50 eleq2 2719 . . . . . 6 (𝑦 = {𝑧} → (𝑧𝑦𝑧 ∈ {𝑧}))
51 ineq2 3841 . . . . . . . . 9 (𝑦 = {𝑧} → (𝑢𝑦) = (𝑢 ∩ {𝑧}))
5251neeq1d 2882 . . . . . . . 8 (𝑦 = {𝑧} → ((𝑢𝑦) ≠ ∅ ↔ (𝑢 ∩ {𝑧}) ≠ ∅))
5352rabbidv 3220 . . . . . . 7 (𝑦 = {𝑧} → {𝑢𝐶 ∣ (𝑢𝑦) ≠ ∅} = {𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅})
5453eleq1d 2715 . . . . . 6 (𝑦 = {𝑧} → ({𝑢𝐶 ∣ (𝑢𝑦) ≠ ∅} ∈ Fin ↔ {𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin))
5550, 54anbi12d 747 . . . . 5 (𝑦 = {𝑧} → ((𝑧𝑦 ∧ {𝑢𝐶 ∣ (𝑢𝑦) ≠ ∅} ∈ Fin) ↔ (𝑧 ∈ {𝑧} ∧ {𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)))
5655rspcev 3340 . . . 4 (({𝑧} ∈ 𝒫 𝑋 ∧ (𝑧 ∈ {𝑧} ∧ {𝑢𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑧𝑦 ∧ {𝑢𝐶 ∣ (𝑢𝑦) ≠ ∅} ∈ Fin))
574, 6, 49, 56syl12anc 1364 . . 3 ((𝑋𝑉𝑧𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑧𝑦 ∧ {𝑢𝐶 ∣ (𝑢𝑦) ≠ ∅} ∈ Fin))
5857ralrimiva 2995 . 2 (𝑋𝑉 → ∀𝑧𝑋𝑦 ∈ 𝒫 𝑋(𝑧𝑦 ∧ {𝑢𝐶 ∣ (𝑢𝑦) ≠ ∅} ∈ Fin))
59 unipw 4948 . . . 4 𝒫 𝑋 = 𝑋
6059eqcomi 2660 . . 3 𝑋 = 𝒫 𝑋
6110unisngl 21378 . . 3 𝑋 = 𝐶
6260, 61islocfin 21368 . 2 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑋 ∧ ∀𝑧𝑋𝑦 ∈ 𝒫 𝑋(𝑧𝑦 ∧ {𝑢𝐶 ∣ (𝑢𝑦) ≠ ∅} ∈ Fin)))
631, 2, 58, 62syl3anbrc 1265 1 (𝑋𝑉𝐶 ∈ (LocFin‘𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  cin 3606  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468  cfv 5926  Fincfn 7997  Topctop 20746  LocFinclocfin 21355
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-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-3or 1055  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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  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-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  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-om 7108  df-1o 7605  df-en 7998  df-fin 8001  df-top 20747  df-locfin 21358
This theorem is referenced by:  dispcmp  30054
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