Theorem topdifinffin 32372

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 3684	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
3	2	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
4	3	notbid 307	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ (𝐴 ∖ 𝑥) ∈ Fin ↔ ¬ (𝐴 ∖ 𝑦) ∈ Fin))
5		eqeq1 2614	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
6		eqeq1 2614	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
7	5, 6	orbi12d 742	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴)))
8	4, 7	orbi12d 742	. . . 4 ⊢ (𝑥 = 𝑦 → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))))
9	8	cbvrabv 3172	. . 3 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
10	1, 9	eqtri 2632	. 2 ⊢ 𝑇 = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
11	10	topdifinffinlem 32371	1 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  ‘cfv 5804  Fincfn 7841  TopOnctopon 20518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-topgen 15927  df-top 20521  df-topon 20523

This theorem is referenced by:  topdifinf  32373