Theorem topdifinf 38540

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 38539	. . 3 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
3	1	topdifinfindis 38537	. . . 4 ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})
4		indistopon 23374	. . . 4 ⊢ (𝐴 ∈ Fin → {∅, 𝐴} ∈ (TopOn‘𝐴))
5	3, 4	eqeltrd 2864	. . 3 ⊢ (𝐴 ∈ Fin → 𝑇 ∈ (TopOn‘𝐴))
6	2, 5	impbii 212	. 2 ⊢ (𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin)
7	2, 3	syl 18	. 2 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴})
8	6, 7	pm3.2i 476	1 ⊢ ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  {crab 3416   ∖ cdif 4048  ∅c0 4281  𝒫 cpw 4560  {cpr 4590  ‘cfv 6566  Fincfn 9011  TopOnctopon 23283

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-om 7912  df-1o 8514  df-en 9012  df-fin 9015  df-topgen 17664  df-top 23267  df-topon 23284

This theorem is used by: (None)