Theorem topdifinfindis 37880

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 1941	. 2 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
2		topdifinf.t	. . 3 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
3		nfrab1 3443	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
4	2, 3	nfcxfr 2929	. 2 ⊢ Ⅎ𝑥𝑇
5		nfcv 2931	. 2 ⊢ Ⅎ𝑥{∅, 𝐴}
6		0elpw 5327	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝐴
7		eleq1a 2864	. . . . . 6 ⊢ (∅ ∈ 𝒫 𝐴 → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
8	6, 7	mp1i 14	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
9		pwidg 4587	. . . . . 6 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴)
10		eleq1a 2864	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 → 𝑥 ∈ 𝒫 𝐴))
11	9, 10	syl 18	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝑥 = 𝐴 → 𝑥 ∈ 𝒫 𝐴))
12	8, 11	jaod 872	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴))
13	12	pm4.71rd 571	. . 3 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
14		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
15	14	elpr 4619	. . . 4 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
16	15	a1i 11	. . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
17	2	reqabi 3446	. . . 4 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
18		diffi 9159	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝑥) ∈ Fin)
19		biortn 950	. . . . . 6 ⊢ ((𝐴 ∖ 𝑥) ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
20	18, 19	syl 18	. . . . 5 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
21	20	anbi2d 641	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))))
22	17, 21	bitr4id 293	. . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
23	13, 16, 22	3bitr4rd 315	. 2 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ {∅, 𝐴}))
24	1, 4, 5, 23	eqrd 3964	1 ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  {crab 3423   ∖ cdif 3910  ∅c0 4294  𝒫 cpw 4567  {cpr 4596  Fincfn 8943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7863  df-1o 8453  df-en 8944  df-fin 8947

This theorem is referenced by:  topdifinf  37883