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Theorem topdifinffin 36880
Description: Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets π‘₯ of 𝐴 such that the complement of π‘₯ in 𝐴 is infinite, or π‘₯ is the empty set, or π‘₯ is all of 𝐴, is a topology only if 𝐴 is finite. (Contributed by ML, 17-Jul-2020.)
Hypothesis
Ref Expression
topdifinf.t 𝑇 = {π‘₯ ∈ 𝒫 𝐴 ∣ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))}
Assertion
Ref Expression
topdifinffin (𝑇 ∈ (TopOnβ€˜π΄) β†’ 𝐴 ∈ Fin)
Distinct variable group:   π‘₯,𝐴
Allowed substitution hint:   𝑇(π‘₯)

Proof of Theorem topdifinffin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 topdifinf.t . . 3 𝑇 = {π‘₯ ∈ 𝒫 𝐴 ∣ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))}
2 difeq2 4109 . . . . . . 7 (π‘₯ = 𝑦 β†’ (𝐴 βˆ– π‘₯) = (𝐴 βˆ– 𝑦))
32eleq1d 2810 . . . . . 6 (π‘₯ = 𝑦 β†’ ((𝐴 βˆ– π‘₯) ∈ Fin ↔ (𝐴 βˆ– 𝑦) ∈ Fin))
43notbid 317 . . . . 5 (π‘₯ = 𝑦 β†’ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ↔ Β¬ (𝐴 βˆ– 𝑦) ∈ Fin))
5 eqeq1 2729 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯ = βˆ… ↔ 𝑦 = βˆ…))
6 eqeq1 2729 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯ = 𝐴 ↔ 𝑦 = 𝐴))
75, 6orbi12d 916 . . . . 5 (π‘₯ = 𝑦 β†’ ((π‘₯ = βˆ… ∨ π‘₯ = 𝐴) ↔ (𝑦 = βˆ… ∨ 𝑦 = 𝐴)))
84, 7orbi12d 916 . . . 4 (π‘₯ = 𝑦 β†’ ((Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴)) ↔ (Β¬ (𝐴 βˆ– 𝑦) ∈ Fin ∨ (𝑦 = βˆ… ∨ 𝑦 = 𝐴))))
98cbvrabv 3430 . . 3 {π‘₯ ∈ 𝒫 𝐴 ∣ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))} = {𝑦 ∈ 𝒫 𝐴 ∣ (Β¬ (𝐴 βˆ– 𝑦) ∈ Fin ∨ (𝑦 = βˆ… ∨ 𝑦 = 𝐴))}
101, 9eqtri 2753 . 2 𝑇 = {𝑦 ∈ 𝒫 𝐴 ∣ (Β¬ (𝐴 βˆ– 𝑦) ∈ Fin ∨ (𝑦 = βˆ… ∨ 𝑦 = 𝐴))}
1110topdifinffinlem 36879 1 (𝑇 ∈ (TopOnβ€˜π΄) β†’ 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∨ wo 845   = wceq 1533   ∈ wcel 2098  {crab 3419   βˆ– cdif 3938  βˆ…c0 4319  π’« cpw 4599  β€˜cfv 6543  Fincfn 8957  TopOnctopon 22825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7866  df-1o 8480  df-en 8958  df-fin 8961  df-topgen 17419  df-top 22809  df-topon 22826
This theorem is referenced by:  topdifinf  36881
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