Theorem topdifinf 35143

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 35142	. . 3 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
3	1	topdifinfindis 35140	. . . 4 ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})
4		indistopon 21752	. . . 4 ⊢ (𝐴 ∈ Fin → {∅, 𝐴} ∈ (TopOn‘𝐴))
5	3, 4	eqeltrd 2833	. . 3 ⊢ (𝐴 ∈ Fin → 𝑇 ∈ (TopOn‘𝐴))
6	2, 5	impbii 212	. 2 ⊢ (𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin)
7	2, 3	syl 17	. 2 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴})
8	6, 7	pm3.2i 474	1 ⊢ ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 846   = wceq 1542   ∈ wcel 2114  {crab 3057   ∖ cdif 3840  ∅c0 4211  𝒫 cpw 4488  {cpr 4518  ‘cfv 6339  Fincfn 8555  TopOnctopon 21661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-1o 8131  df-en 8556  df-fin 8559  df-topgen 16820  df-top 21645  df-topon 21662

This theorem is referenced by: (None)