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Theorem topdifinffin 33507
 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 3865 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
32eleq1d 2824 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
43notbid 307 . . . . 5 (𝑥 = 𝑦 → (¬ (𝐴𝑥) ∈ Fin ↔ ¬ (𝐴𝑦) ∈ Fin))
5 eqeq1 2764 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
6 eqeq1 2764 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
75, 6orbi12d 748 . . . . 5 (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴)))
84, 7orbi12d 748 . . . 4 (𝑥 = 𝑦 → ((¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))))
98cbvrabv 3339 . . 3 {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
101, 9eqtri 2782 . 2 𝑇 = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
1110topdifinffinlem 33506 1 (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   = wceq 1632   ∈ wcel 2139  {crab 3054   ∖ cdif 3712  ∅c0 4058  𝒫 cpw 4302  ‘cfv 6049  Fincfn 8121  TopOnctopon 20917 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-topgen 16306  df-top 20901  df-topon 20918 This theorem is referenced by:  topdifinf  33508
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