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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 21547. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 21549 and indistps2ALT 21550 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 5202 | . . . 4 ⊢ ∅ ∈ V | |
4 | fvex 6676 | . . . . 5 ⊢ (Base‘𝐾) ∈ V | |
5 | 1, 4 | eqeltrri 2907 | . . . 4 ⊢ 𝐴 ∈ V |
6 | 3, 5 | unipr 4843 | . . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴) |
7 | uncom 4126 | . . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
8 | un0 4341 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
9 | 6, 7, 8 | 3eqtrri 2846 | . 2 ⊢ 𝐴 = ∪ {∅, 𝐴} |
10 | indistop 21538 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
11 | 1, 2, 9, 10 | istpsi 21478 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ∅c0 4288 {cpr 4559 ∪ cuni 4830 ‘cfv 6348 Basecbs 16471 TopOpenctopn 16683 TopSpctps 21468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-top 21430 df-topon 21447 df-topsp 21469 |
This theorem is referenced by: (None) |
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