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Mirrors > Home > MPE Home > Th. List > Mathboxes > topdifinfindis | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
topdifinf.t | ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} |
Ref | Expression |
---|---|
topdifinfindis | ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . 2 ⊢ Ⅎ𝑥 𝐴 ∈ Fin | |
2 | topdifinf.t | . . 3 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} | |
3 | nfrab1 3122 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} | |
4 | 2, 3 | nfcxfr 2762 | . 2 ⊢ Ⅎ𝑥𝑇 |
5 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥{∅, 𝐴} | |
6 | 0elpw 4834 | . . . . . 6 ⊢ ∅ ∈ 𝒫 𝐴 | |
7 | eleq1a 2696 | . . . . . 6 ⊢ (∅ ∈ 𝒫 𝐴 → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴)) | |
8 | 6, 7 | mp1i 13 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴)) |
9 | pwidg 4173 | . . . . . 6 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴) | |
10 | eleq1a 2696 | . . . . . 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 395 | . . . 4 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴)) |
13 | 12 | pm4.71rd 667 | . . 3 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))) |
14 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
15 | 14 | elpr 4198 | . . . 4 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))) |
17 | diffi 8192 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝑥) ∈ Fin) | |
18 | biortn 421 | . . . . . 6 ⊢ ((𝐴 ∖ 𝑥) ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))) |
20 | 19 | anbi2d 740 | . . . 4 ⊢ (𝐴 ∈ Fin → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))) |
21 | 2 | rabeq2i 3197 | . . . 4 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))) |
22 | 20, 21 | syl6rbbr 279 | . . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))) |
23 | 13, 16, 22 | 3bitr4rd 301 | . 2 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ {∅, 𝐴})) |
24 | 1, 4, 5, 23 | eqrd 3622 | 1 ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 ∖ cdif 3571 ∅c0 3915 𝒫 cpw 4158 {cpr 4179 Fincfn 7955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-om 7066 df-er 7742 df-en 7956 df-fin 7959 |
This theorem is referenced by: topdifinf 33197 |
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