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| Mirrors > Home > MPE Home > Th. List > indistps2 | Structured version Visualization version GIF version | ||
| Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 23018. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 23020 and indistps2ALT 23022 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.) | 
| Ref | Expression | 
|---|---|
| indistps2.a | ⊢ (Base‘𝐾) = 𝐴 | 
| indistps2.j | ⊢ (TopOpen‘𝐾) = {∅, 𝐴} | 
| Ref | Expression | 
|---|---|
| indistps2 | ⊢ 𝐾 ∈ TopSp | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | indistps2.a | . 2 ⊢ (Base‘𝐾) = 𝐴 | |
| 2 | indistps2.j | . 2 ⊢ (TopOpen‘𝐾) = {∅, 𝐴} | |
| 3 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 4 | fvex 6919 | . . . . 5 ⊢ (Base‘𝐾) ∈ V | |
| 5 | 1, 4 | eqeltrri 2838 | . . . 4 ⊢ 𝐴 ∈ V | 
| 6 | 3, 5 | unipr 4924 | . . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴) | 
| 7 | uncom 4158 | . . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
| 8 | un0 4394 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 9 | 6, 7, 8 | 3eqtrri 2770 | . 2 ⊢ 𝐴 = ∪ {∅, 𝐴} | 
| 10 | indistop 23009 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
| 11 | 1, 2, 9, 10 | istpsi 22948 | 1 ⊢ 𝐾 ∈ TopSp | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∅c0 4333 {cpr 4628 ∪ cuni 4907 ‘cfv 6561 Basecbs 17247 TopOpenctopn 17466 TopSpctps 22938 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-top 22900 df-topon 22917 df-topsp 22939 | 
| This theorem is referenced by: (None) | 
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