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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 1915 | . 2 ⊢ Ⅎ𝑥 𝐴 ∈ Fin | |
2 | topdifinf.t | . . 3 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} | |
3 | nfrab1 3337 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} | |
4 | 2, 3 | nfcxfr 2953 | . 2 ⊢ Ⅎ𝑥𝑇 |
5 | nfcv 2955 | . 2 ⊢ Ⅎ𝑥{∅, 𝐴} | |
6 | 0elpw 5221 | . . . . . 6 ⊢ ∅ ∈ 𝒫 𝐴 | |
7 | eleq1a 2885 | . . . . . 6 ⊢ (∅ ∈ 𝒫 𝐴 → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴)) | |
8 | 6, 7 | mp1i 13 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴)) |
9 | pwidg 4519 | . . . . . 6 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴) | |
10 | eleq1a 2885 | . . . . . 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 856 | . . . 4 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴)) |
13 | 12 | pm4.71rd 566 | . . 3 ⊢ (𝐴 ∈ Fin → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))) |
14 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
15 | 14 | elpr 4548 | . . . 4 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))) |
17 | 2 | rabeq2i 3435 | . . . 4 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))) |
18 | diffi 8734 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝑥) ∈ Fin) | |
19 | biortn 935 | . . . . . 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 631 | . . . 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 3934 | 1 ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 {crab 3110 ∖ cdif 3878 ∅c0 4243 𝒫 cpw 4497 {cpr 4527 Fincfn 8492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-om 7561 df-er 8272 df-en 8493 df-fin 8496 |
This theorem is referenced by: topdifinf 34766 |
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