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Mirrors > Home > MPE Home > Th. List > indistps2ALT | Structured version Visualization version GIF version |
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 21766 from the structural version indistps 21765. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
indistps2ALT.a | ⊢ (Base‘𝐾) = 𝐴 |
indistps2ALT.j | ⊢ (TopOpen‘𝐾) = {∅, 𝐴} |
Ref | Expression |
---|---|
indistps2ALT | ⊢ 𝐾 ∈ TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indistps2ALT.a | . . . 4 ⊢ (Base‘𝐾) = 𝐴 | |
2 | fvex 6690 | . . . 4 ⊢ (Base‘𝐾) ∈ V | |
3 | 1, 2 | eqeltrri 2831 | . . 3 ⊢ 𝐴 ∈ V |
4 | indistopon 21755 | . . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴) |
6 | 1 | eqcomi 2748 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
7 | indistps2ALT.j | . . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴} | |
8 | 7 | eqcomi 2748 | . . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾) |
9 | 6, 8 | istps 21688 | . 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴)) |
10 | 5, 9 | mpbir 234 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 Vcvv 3399 ∅c0 4212 {cpr 4519 ‘cfv 6340 Basecbs 16589 TopOpenctopn 16801 TopOnctopon 21664 TopSpctps 21686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-iota 6298 df-fun 6342 df-fv 6348 df-top 21648 df-topon 21665 df-topsp 21687 |
This theorem is referenced by: (None) |
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