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Theorem indistps2ALT 23038
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 23035 from the structural version indistps 23034. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
indistps2ALT.a (Base‘𝐾) = 𝐴
indistps2ALT.j (TopOpen‘𝐾) = {∅, 𝐴}
Assertion
Ref Expression
indistps2ALT 𝐾 ∈ TopSp

Proof of Theorem indistps2ALT
StepHypRef Expression
1 indistps2ALT.a . . . 4 (Base‘𝐾) = 𝐴
2 fvex 6920 . . . 4 (Base‘𝐾) ∈ V
31, 2eqeltrri 2836 . . 3 𝐴 ∈ V
4 indistopon 23024 . . 3 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
53, 4ax-mp 5 . 2 {∅, 𝐴} ∈ (TopOn‘𝐴)
61eqcomi 2744 . . 3 𝐴 = (Base‘𝐾)
7 indistps2ALT.j . . . 4 (TopOpen‘𝐾) = {∅, 𝐴}
87eqcomi 2744 . . 3 {∅, 𝐴} = (TopOpen‘𝐾)
96, 8istps 22956 . 2 (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴))
105, 9mpbir 231 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {cpr 4633  cfv 6563  Basecbs 17245  TopOpenctopn 17468  TopOnctopon 22932  TopSpctps 22954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-top 22916  df-topon 22933  df-topsp 22955
This theorem is referenced by: (None)
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