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Theorem indistps2 20864
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 20863. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 20865 and indistps2ALT 20866 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 4823 . . . 4 ∅ ∈ V
4 fvex 6239 . . . . 5 (Base‘𝐾) ∈ V
51, 4eqeltrri 2727 . . . 4 𝐴 ∈ V
63, 5unipr 4481 . . 3 {∅, 𝐴} = (∅ ∪ 𝐴)
7 uncom 3790 . . 3 (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8 un0 4000 . . 3 (𝐴 ∪ ∅) = 𝐴
96, 7, 83eqtrri 2678 . 2 𝐴 = {∅, 𝐴}
10 indistop 20854 . 2 {∅, 𝐴} ∈ Top
111, 2, 9, 10istpsi 20794 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  c0 3948  {cpr 4212   cuni 4468  cfv 5926  Basecbs 15904  TopOpenctopn 16129  TopSpctps 20784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-top 20747  df-topon 20764  df-topsp 20785
This theorem is referenced by: (None)
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