Theorem topdifinf 37554

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 if and only if 𝐴 is finite, in which case it is the trivial topology. (Contributed by ML, 17-Jul-2020.)

Hypothesis

Ref	Expression
topdifinf.t	⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}

Assertion

Ref	Expression
topdifinf	⊢ ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴}))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem topdifinf

Step	Hyp	Ref	Expression
1		topdifinf.t	. . . 4 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
2	1	topdifinffin 37553	. . 3 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
3	1	topdifinfindis 37551	. . . 4 ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})
4		indistopon 22945	. . . 4 ⊢ (𝐴 ∈ Fin → {∅, 𝐴} ∈ (TopOn‘𝐴))
5	3, 4	eqeltrd 2836	. . 3 ⊢ (𝐴 ∈ Fin → 𝑇 ∈ (TopOn‘𝐴))
6	2, 5	impbii 209	. 2 ⊢ (𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin)
7	2, 3	syl 17	. 2 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴})
8	6, 7	pm3.2i 470	1 ⊢ ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  {crab 3399   ∖ cdif 3898  ∅c0 4285  𝒫 cpw 4554  {cpr 4582  ‘cfv 6492  Fincfn 8883  TopOnctopon 22854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-en 8884  df-fin 8887  df-topgen 17363  df-top 22838  df-topon 22855

This theorem is referenced by: (None)