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Theorem topdifinffin 34646
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 a topology only if 𝐴 is finite. (Contributed by ML, 17-Jul-2020.)
Hypothesis
Ref Expression
topdifinf.t 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
Assertion
Ref Expression
topdifinffin (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem topdifinffin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 topdifinf.t . . 3 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
2 difeq2 4072 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
32eleq1d 2895 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
43notbid 320 . . . . 5 (𝑥 = 𝑦 → (¬ (𝐴𝑥) ∈ Fin ↔ ¬ (𝐴𝑦) ∈ Fin))
5 eqeq1 2824 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
6 eqeq1 2824 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
75, 6orbi12d 915 . . . . 5 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴)))
84, 7orbi12d 915 . . . 4 (𝑥 = 𝑦 → ((¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))))
98cbvrabv 3470 . . 3 {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
101, 9eqtri 2843 . 2 𝑇 = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
1110topdifinffinlem 34645 1 (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 843   = wceq 1537  wcel 2114  {crab 3129  cdif 3910  c0 4269  𝒫 cpw 4515  cfv 6331  Fincfn 8487  TopOnctopon 21494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-en 8488  df-fin 8491  df-topgen 16696  df-top 21478  df-topon 21495
This theorem is referenced by:  topdifinf  34647
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