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Mirrors > Home > MPE Home > Th. List > Mathboxes > topdifinffin | 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 a topology only if 𝐴 is finite. (Contributed by ML, 17-Jul-2020.) |
Ref | Expression |
---|---|
topdifinf.t | ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} |
Ref | Expression |
---|---|
topdifinffin | ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topdifinf.t | . . 3 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} | |
2 | difeq2 4044 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦)) | |
3 | 2 | eleq1d 2874 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin)) |
4 | 3 | notbid 321 | . . . . 5 ⊢ (𝑥 = 𝑦 → (¬ (𝐴 ∖ 𝑥) ∈ Fin ↔ ¬ (𝐴 ∖ 𝑦) ∈ Fin)) |
5 | eqeq1 2802 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
6 | eqeq1 2802 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
7 | 5, 6 | orbi12d 916 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴))) |
8 | 4, 7 | orbi12d 916 | . . . 4 ⊢ (𝑥 = 𝑦 → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴)))) |
9 | 8 | cbvrabv 3439 | . . 3 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))} |
10 | 1, 9 | eqtri 2821 | . 2 ⊢ 𝑇 = {𝑦 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑦) ∈ Fin ∨ (𝑦 = ∅ ∨ 𝑦 = 𝐴))} |
11 | 10 | topdifinffinlem 34764 | 1 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 844 = wceq 1538 ∈ wcel 2111 {crab 3110 ∖ cdif 3878 ∅c0 4243 𝒫 cpw 4497 ‘cfv 6324 Fincfn 8492 TopOnctopon 21515 |
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-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 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-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 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-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-fin 8496 df-topgen 16709 df-top 21499 df-topon 21516 |
This theorem is referenced by: topdifinf 34766 |
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