Theorem topdifinfindis 38012

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 1944	. 2 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
2		topdifinf.t	. . 3 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
3		nfrab1 3436	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
4	2, 3	nfcxfr 2923	. 2 ⊢ Ⅎ𝑥𝑇
5		nfcv 2925	. 2 ⊢ Ⅎ𝑥{∅, 𝐴}
6		0elpw 5326	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝐴
7		eleq1a 2858	. . . . . 6 ⊢ (∅ ∈ 𝒫 𝐴 → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
8	6, 7	mp1i 14	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴))
9		pwidg 4582	. . . . . 6 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴)
10		eleq1a 2858	. . . . . 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 3459	. . . . 5 ⊢ 𝑥 ∈ V
15	14	elpr 4614	. . . 4 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
16	15	a1i 11	. . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
17	2	reqabi 3439	. . . 4 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
18		diffi 9155	. . . . . 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 3956	1 ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  {crab 3416   ∖ cdif 3902  ∅c0 4286  𝒫 cpw 4562  {cpr 4591  Fincfn 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7859  df-1o 8449  df-en 8940  df-fin 8943

This theorem is referenced by:  topdifinf  38015