Theorem topdifinffin 37917

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 ∣ (¬ (𝐴 ∖ 𝑥) ∈ 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.

Step	Hyp	Ref	Expression
1		topdifinf.t	. . 3 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
2		difeq2 4081	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
3	2	eleq1d 2854	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
4	3	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ (𝐴 ∖ 𝑥) ∈ Fin ↔ ¬ (𝐴 ∖ 𝑦) ∈ Fin))
5		eqeq1 2773	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
6		eqeq1 2773	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
7	5, 6	orbi12d 931	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴)))
8	4, 7	orbi12d 931	. . . 4 ⊢ (𝑥 = 𝑦 → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))))
9	8	cbvrabv 3432	. . 3 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
10	1, 9	eqtri 2792	. 2 ⊢ 𝑇 = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
11	10	topdifinffinlem 37916	1 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 860   = wceq 1567   ∈ wcel 2149  {crab 3422   ∖ cdif 3908  ∅c0 4292  𝒫 cpw 4565  ‘cfv 6537  Fincfn 8943  TopOnctopon 23036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7863  df-1o 8453  df-en 8944  df-fin 8947  df-topgen 17496  df-top 23020  df-topon 23037

This theorem is referenced by:  topdifinf  37918