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Theorem topdifinfindis 32169
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 1828 . 2 𝑥 𝐴 ∈ Fin
2 topdifinf.t . . 3 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
3 nfrab1 3094 . . 3 𝑥{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
42, 3nfcxfr 2744 . 2 𝑥𝑇
5 nfcv 2746 . 2 𝑥{∅, 𝐴}
6 0elpw 4751 . . . . . 6 ∅ ∈ 𝒫 𝐴
7 eleq1a 2678 . . . . . 6 (∅ ∈ 𝒫 𝐴 → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
86, 7mp1i 13 . . . . 5 (𝐴 ∈ Fin → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
9 pwidg 4116 . . . . . 6 (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴)
10 eleq1a 2678 . . . . . 6 (𝐴 ∈ 𝒫 𝐴 → (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴))
119, 10syl 17 . . . . 5 (𝐴 ∈ Fin → (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴))
128, 11jaod 393 . . . 4 (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴))
1312pm4.71rd 664 . . 3 (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
14 vex 3171 . . . . 5 𝑥 ∈ V
1514elpr 4141 . . . 4 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
1615a1i 11 . . 3 (𝐴 ∈ Fin → (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
17 diffi 8050 . . . . . 6 (𝐴 ∈ Fin → (𝐴𝑥) ∈ Fin)
18 biortn 419 . . . . . 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 735 . . . 4 (𝐴 ∈ Fin → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))))
212rabeq2i 3165 . . . 4 (𝑥𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2220, 21syl6rbbr 277 . . 3 (𝐴 ∈ Fin → (𝑥𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2313, 16, 223bitr4rd 299 . 2 (𝐴 ∈ Fin → (𝑥𝑇𝑥 ∈ {∅, 𝐴}))
241, 4, 5, 23eqrd 3581 1 (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1975  {crab 2895  cdif 3532  c0 3869  𝒫 cpw 4103  {cpr 4122  Fincfn 7814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-om 6931  df-er 7602  df-en 7815  df-fin 7818
This theorem is referenced by:  topdifinf  32172
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