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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 21193. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 21195 and indistps2ALT 21196 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 5016 | . . . 4 ⊢ ∅ ∈ V | |
4 | fvex 6450 | . . . . 5 ⊢ (Base‘𝐾) ∈ V | |
5 | 1, 4 | eqeltrri 2903 | . . . 4 ⊢ 𝐴 ∈ V |
6 | 3, 5 | unipr 4673 | . . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴) |
7 | uncom 3986 | . . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
8 | un0 4194 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
9 | 6, 7, 8 | 3eqtrri 2854 | . 2 ⊢ 𝐴 = ∪ {∅, 𝐴} |
10 | indistop 21184 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
11 | 1, 2, 9, 10 | istpsi 21124 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∪ cun 3796 ∅c0 4146 {cpr 4401 ∪ cuni 4660 ‘cfv 6127 Basecbs 16229 TopOpenctopn 16442 TopSpctps 21114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-top 21076 df-topon 21093 df-topsp 21115 |
This theorem is referenced by: (None) |
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