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Theorem topdifinfindis 33324
Description: Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is the trivial topology when 𝐴 is finite. (Contributed by ML, 14-Jul-2020.)
Hypothesis
Ref Expression
topdifinf.t 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
Assertion
Ref Expression
topdifinfindis (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem topdifinfindis
StepHypRef Expression
1 nfv 1883 . 2 𝑥 𝐴 ∈ Fin
2 topdifinf.t . . 3 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
3 nfrab1 3152 . . 3 𝑥{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
42, 3nfcxfr 2791 . 2 𝑥𝑇
5 nfcv 2793 . 2 𝑥{∅, 𝐴}
6 0elpw 4864 . . . . . 6 ∅ ∈ 𝒫 𝐴
7 eleq1a 2725 . . . . . 6 (∅ ∈ 𝒫 𝐴 → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
86, 7mp1i 13 . . . . 5 (𝐴 ∈ Fin → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
9 pwidg 4206 . . . . . 6 (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴)
10 eleq1a 2725 . . . . . 6 (𝐴 ∈ 𝒫 𝐴 → (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴))
119, 10syl 17 . . . . 5 (𝐴 ∈ Fin → (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴))
128, 11jaod 394 . . . 4 (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴))
1312pm4.71rd 668 . . 3 (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
14 vex 3234 . . . . 5 𝑥 ∈ V
1514elpr 4231 . . . 4 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
1615a1i 11 . . 3 (𝐴 ∈ Fin → (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
17 diffi 8233 . . . . . 6 (𝐴 ∈ Fin → (𝐴𝑥) ∈ Fin)
18 biortn 420 . . . . . 6 ((𝐴𝑥) ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
1917, 18syl 17 . . . . 5 (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2019anbi2d 740 . . . 4 (𝐴 ∈ Fin → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))))
212rabeq2i 3228 . . . 4 (𝑥𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2220, 21syl6rbbr 279 . . 3 (𝐴 ∈ Fin → (𝑥𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2313, 16, 223bitr4rd 301 . 2 (𝐴 ∈ Fin → (𝑥𝑇𝑥 ∈ {∅, 𝐴}))
241, 4, 5, 23eqrd 3655 1 (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  {crab 2945  cdif 3604  c0 3948  𝒫 cpw 4191  {cpr 4212  Fincfn 7997
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-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-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-om 7108  df-er 7787  df-en 7998  df-fin 8001
This theorem is referenced by:  topdifinf  33327
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