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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 22378 from the structural version indistps 22377. (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 6856 | . . . 4 ⊢ (Base‘𝐾) ∈ V | |
3 | 1, 2 | eqeltrri 2831 | . . 3 ⊢ 𝐴 ∈ V |
4 | indistopon 22367 | . . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴) |
6 | 1 | eqcomi 2742 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
7 | indistps2ALT.j | . . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴} | |
8 | 7 | eqcomi 2742 | . . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾) |
9 | 6, 8 | istps 22299 | . 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴)) |
10 | 5, 9 | mpbir 230 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 {cpr 4589 ‘cfv 6497 Basecbs 17088 TopOpenctopn 17308 TopOnctopon 22275 TopSpctps 22297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-top 22259 df-topon 22276 df-topsp 22298 |
This theorem is referenced by: (None) |
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