Theorem topdifinfindis 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 the trivial topology when 𝐴 is finite. (Contributed by ML, 14-Jul-2020.)

Hypothesis

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

Assertion

Ref	Expression
topdifinfindis	⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem topdifinfindis

Step	Hyp	Ref	Expression
1		nfv 1912	. 2 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
2		topdifinf.t	. . 3 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
3		nfrab1 3454	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
4	2, 3	nfcxfr 2901	. 2 ⊢ Ⅎ𝑥𝑇
5		nfcv 2903	. 2 ⊢ Ⅎ𝑥{∅, 𝐴}
6		0elpw 5362	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝐴
7		eleq1a 2834	. . . . . 6 ⊢ (∅ ∈ 𝒫 𝐴 → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
8	6, 7	mp1i 13	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
9		pwidg 4625	. . . . . 6 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴)
10		eleq1a 2834	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 → 𝑥 ∈ 𝒫 𝐴))
11	9, 10	syl 17	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝑥 = 𝐴 → 𝑥 ∈ 𝒫 𝐴))
12	8, 11	jaod 859	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴))
13	12	pm4.71rd 562	. . 3 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
14		vex 3482	. . . . 5 ⊢ 𝑥 ∈ V
15	14	elpr 4655	. . . 4 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
16	15	a1i 11	. . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
17	2	reqabi 3457	. . . 4 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
18		diffi 9214	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝑥) ∈ Fin)
19		biortn 937	. . . . . 6 ⊢ ((𝐴 ∖ 𝑥) ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
20	18, 19	syl 17	. . . . 5 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
21	20	anbi2d 630	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))))
22	17, 21	bitr4id 290	. . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
23	13, 16, 22	3bitr4rd 312	. 2 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ {∅, 𝐴}))
24	1, 4, 5, 23	eqrd 4015	1 ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106  {crab 3433   ∖ cdif 3960  ∅c0 4339  𝒫 cpw 4605  {cpr 4633  Fincfn 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-fin 8988

This theorem is referenced by:  topdifinf  37332