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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 22446 from the structural version indistps 22445. (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 6892 | . . . 4 ⊢ (Base‘𝐾) ∈ V | |
3 | 1, 2 | eqeltrri 2830 | . . 3 ⊢ 𝐴 ∈ V |
4 | indistopon 22435 | . . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴) |
6 | 1 | eqcomi 2741 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
7 | indistps2ALT.j | . . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴} | |
8 | 7 | eqcomi 2741 | . . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾) |
9 | 6, 8 | istps 22367 | . 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴)) |
10 | 5, 9 | mpbir 230 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4319 {cpr 4625 ‘cfv 6533 Basecbs 17128 TopOpenctopn 17351 TopOnctopon 22343 TopSpctps 22365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-iota 6485 df-fun 6535 df-fv 6541 df-top 22327 df-topon 22344 df-topsp 22366 |
This theorem is referenced by: (None) |
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