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Mirrors > Home > MPE Home > Th. List > indistps2ALT | Structured version Visualization version GIF version |
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 23035 from the structural version indistps 23034. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
indistps2ALT.a | ⊢ (Base‘𝐾) = 𝐴 |
indistps2ALT.j | ⊢ (TopOpen‘𝐾) = {∅, 𝐴} |
Ref | Expression |
---|---|
indistps2ALT | ⊢ 𝐾 ∈ TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indistps2ALT.a | . . . 4 ⊢ (Base‘𝐾) = 𝐴 | |
2 | fvex 6920 | . . . 4 ⊢ (Base‘𝐾) ∈ V | |
3 | 1, 2 | eqeltrri 2836 | . . 3 ⊢ 𝐴 ∈ V |
4 | indistopon 23024 | . . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴) |
6 | 1 | eqcomi 2744 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
7 | indistps2ALT.j | . . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴} | |
8 | 7 | eqcomi 2744 | . . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾) |
9 | 6, 8 | istps 22956 | . 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴)) |
10 | 5, 9 | mpbir 231 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {cpr 4633 ‘cfv 6563 Basecbs 17245 TopOpenctopn 17468 TopOnctopon 22932 TopSpctps 22954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-top 22916 df-topon 22933 df-topsp 22955 |
This theorem is referenced by: (None) |
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