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Theorem indistps2ALT 23023
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 23020 from the structural version indistps 23019. (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 6918 . . . 4 (Base‘𝐾) ∈ V
31, 2eqeltrri 2837 . . 3 𝐴 ∈ V
4 indistopon 23009 . . 3 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
53, 4ax-mp 5 . 2 {∅, 𝐴} ∈ (TopOn‘𝐴)
61eqcomi 2745 . . 3 𝐴 = (Base‘𝐾)
7 indistps2ALT.j . . . 4 (TopOpen‘𝐾) = {∅, 𝐴}
87eqcomi 2745 . . 3 {∅, 𝐴} = (TopOpen‘𝐾)
96, 8istps 22941 . 2 (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴))
105, 9mpbir 231 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  Vcvv 3479  c0 4332  {cpr 4627  cfv 6560  Basecbs 17248  TopOpenctopn 17467  TopOnctopon 22917  TopSpctps 22939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-top 22901  df-topon 22918  df-topsp 22940
This theorem is referenced by: (None)
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