Theorem topdifinffin 37329

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 4079	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
3	2	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
4	3	notbid 318	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ (𝐴 ∖ 𝑥) ∈ Fin ↔ ¬ (𝐴 ∖ 𝑦) ∈ Fin))
5		eqeq1 2733	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
6		eqeq1 2733	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
7	5, 6	orbi12d 918	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴)))
8	4, 7	orbi12d 918	. . . 4 ⊢ (𝑥 = 𝑦 → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))))
9	8	cbvrabv 3413	. . 3 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
10	1, 9	eqtri 2752	. 2 ⊢ 𝑇 = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))}
11	10	topdifinffinlem 37328	1 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {crab 3402   ∖ cdif 3908  ∅c0 4292  𝒫 cpw 4559  ‘cfv 6499  Fincfn 8895  TopOnctopon 22830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-en 8896  df-fin 8899  df-topgen 17382  df-top 22814  df-topon 22831

This theorem is referenced by:  topdifinf  37330