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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 22927. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22929 and indistps2ALT 22931 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 6910 | . . . . 5 ⊢ (Base‘𝐾) ∈ V | |
5 | 1, 4 | eqeltrri 2826 | . . . 4 ⊢ 𝐴 ∈ V |
6 | 3, 5 | unipr 4925 | . . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴) |
7 | uncom 4152 | . . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
8 | un0 4391 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
9 | 6, 7, 8 | 3eqtrri 2761 | . 2 ⊢ 𝐴 = ∪ {∅, 𝐴} |
10 | indistop 22918 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
11 | 1, 2, 9, 10 | istpsi 22857 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∪ cun 3945 ∅c0 4323 {cpr 4631 ∪ cuni 4908 ‘cfv 6548 Basecbs 17180 TopOpenctopn 17403 TopSpctps 22847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-top 22809 df-topon 22826 df-topsp 22848 |
This theorem is referenced by: (None) |
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