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Theorem topdifinfindis 35043
 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 1915 . 2 𝑥 𝐴 ∈ Fin
2 topdifinf.t . . 3 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
3 nfrab1 3302 . . 3 𝑥{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
42, 3nfcxfr 2917 . 2 𝑥𝑇
5 nfcv 2919 . 2 𝑥{∅, 𝐴}
6 0elpw 5224 . . . . . 6 ∅ ∈ 𝒫 𝐴
7 eleq1a 2847 . . . . . 6 (∅ ∈ 𝒫 𝐴 → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
86, 7mp1i 13 . . . . 5 (𝐴 ∈ Fin → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
9 pwidg 4516 . . . . . 6 (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴)
10 eleq1a 2847 . . . . . 6 (𝐴 ∈ 𝒫 𝐴 → (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴))
119, 10syl 17 . . . . 5 (𝐴 ∈ Fin → (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴))
128, 11jaod 856 . . . 4 (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴))
1312pm4.71rd 566 . . 3 (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
14 vex 3413 . . . . 5 𝑥 ∈ V
1514elpr 4545 . . . 4 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
1615a1i 11 . . 3 (𝐴 ∈ Fin → (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
172rabeq2i 3400 . . . 4 (𝑥𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
18 diffi 8786 . . . . . 6 (𝐴 ∈ Fin → (𝐴𝑥) ∈ Fin)
19 biortn 935 . . . . . 6 ((𝐴𝑥) ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2018, 19syl 17 . . . . 5 (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2120anbi2d 631 . . . 4 (𝐴 ∈ Fin → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))))
2217, 21bitr4id 293 . . 3 (𝐴 ∈ Fin → (𝑥𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2313, 16, 223bitr4rd 315 . 2 (𝐴 ∈ Fin → (𝑥𝑇𝑥 ∈ {∅, 𝐴}))
241, 4, 5, 23eqrd 3911 1 (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {crab 3074   ∖ cdif 3855  ∅c0 4225  𝒫 cpw 4494  {cpr 4524  Fincfn 8527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-1o 8112  df-en 8528  df-fin 8531 This theorem is referenced by:  topdifinf  35046
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