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Theorem indistps2 21018
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 21017. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 21019 and indistps2ALT 21020 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)
Hypotheses
Ref Expression
indistps2.a (Base‘𝐾) = 𝐴
indistps2.j (TopOpen‘𝐾) = {∅, 𝐴}
Assertion
Ref Expression
indistps2 𝐾 ∈ TopSp

Proof of Theorem indistps2
StepHypRef Expression
1 indistps2.a . 2 (Base‘𝐾) = 𝐴
2 indistps2.j . 2 (TopOpen‘𝐾) = {∅, 𝐴}
3 0ex 4942 . . . 4 ∅ ∈ V
4 fvex 6362 . . . . 5 (Base‘𝐾) ∈ V
51, 4eqeltrri 2836 . . . 4 𝐴 ∈ V
63, 5unipr 4601 . . 3 {∅, 𝐴} = (∅ ∪ 𝐴)
7 uncom 3900 . . 3 (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8 un0 4110 . . 3 (𝐴 ∪ ∅) = 𝐴
96, 7, 83eqtrri 2787 . 2 𝐴 = {∅, 𝐴}
10 indistop 21008 . 2 {∅, 𝐴} ∈ Top
111, 2, 9, 10istpsi 20948 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  c0 4058  {cpr 4323   cuni 4588  cfv 6049  Basecbs 16059  TopOpenctopn 16284  TopSpctps 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-top 20901  df-topon 20918  df-topsp 20939
This theorem is referenced by: (None)
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