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Theorem topdifinffin 36750
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 4112 . . . . . . 7 (π‘₯ = 𝑦 β†’ (𝐴 βˆ– π‘₯) = (𝐴 βˆ– 𝑦))
32eleq1d 2813 . . . . . 6 (π‘₯ = 𝑦 β†’ ((𝐴 βˆ– π‘₯) ∈ Fin ↔ (𝐴 βˆ– 𝑦) ∈ Fin))
43notbid 318 . . . . 5 (π‘₯ = 𝑦 β†’ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ↔ Β¬ (𝐴 βˆ– 𝑦) ∈ Fin))
5 eqeq1 2731 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯ = βˆ… ↔ 𝑦 = βˆ…))
6 eqeq1 2731 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯ = 𝐴 ↔ 𝑦 = 𝐴))
75, 6orbi12d 917 . . . . 5 (π‘₯ = 𝑦 β†’ ((π‘₯ = βˆ… ∨ π‘₯ = 𝐴) ↔ (𝑦 = βˆ… ∨ 𝑦 = 𝐴)))
84, 7orbi12d 917 . . . 4 (π‘₯ = 𝑦 β†’ ((Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴)) ↔ (Β¬ (𝐴 βˆ– 𝑦) ∈ Fin ∨ (𝑦 = βˆ… ∨ 𝑦 = 𝐴))))
98cbvrabv 3437 . . 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 2755 . 2 𝑇 = {𝑦 ∈ 𝒫 𝐴 ∣ (Β¬ (𝐴 βˆ– 𝑦) ∈ Fin ∨ (𝑦 = βˆ… ∨ 𝑦 = 𝐴))}
1110topdifinffinlem 36749 1 (𝑇 ∈ (TopOnβ€˜π΄) β†’ 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∨ wo 846   = wceq 1534   ∈ wcel 2099  {crab 3427   βˆ– cdif 3941  βˆ…c0 4318  π’« cpw 4598  β€˜cfv 6542  Fincfn 8953  TopOnctopon 22786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7863  df-1o 8478  df-en 8954  df-fin 8957  df-topgen 17410  df-top 22770  df-topon 22787
This theorem is referenced by:  topdifinf  36751
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