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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 23019. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 23021 and indistps2ALT 23023 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 5306 | . . . 4 ⊢ ∅ ∈ V | |
| 4 | fvex 6918 | . . . . 5 ⊢ (Base‘𝐾) ∈ V | |
| 5 | 1, 4 | eqeltrri 2837 | . . . 4 ⊢ 𝐴 ∈ V | 
| 6 | 3, 5 | unipr 4923 | . . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴) | 
| 7 | uncom 4157 | . . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
| 8 | un0 4393 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 9 | 6, 7, 8 | 3eqtrri 2769 | . 2 ⊢ 𝐴 = ∪ {∅, 𝐴} | 
| 10 | indistop 23010 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
| 11 | 1, 2, 9, 10 | istpsi 22949 | 1 ⊢ 𝐾 ∈ TopSp | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ∅c0 4332 {cpr 4627 ∪ cuni 4906 ‘cfv 6560 Basecbs 17248 TopOpenctopn 17467 TopSpctps 22939 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-top 22901 df-topon 22918 df-topsp 22940 | 
| This theorem is referenced by: (None) | 
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