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Theorem indistps2 21548
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 21547. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 21549 and indistps2ALT 21550 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 5202 . . . 4 ∅ ∈ V
4 fvex 6676 . . . . 5 (Base‘𝐾) ∈ V
51, 4eqeltrri 2907 . . . 4 𝐴 ∈ V
63, 5unipr 4843 . . 3 {∅, 𝐴} = (∅ ∪ 𝐴)
7 uncom 4126 . . 3 (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8 un0 4341 . . 3 (𝐴 ∪ ∅) = 𝐴
96, 7, 83eqtrri 2846 . 2 𝐴 = {∅, 𝐴}
10 indistop 21538 . 2 {∅, 𝐴} ∈ Top
111, 2, 9, 10istpsi 21478 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  c0 4288  {cpr 4559   cuni 4830  cfv 6348  Basecbs 16471  TopOpenctopn 16683  TopSpctps 21468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-topon 21447  df-topsp 21469
This theorem is referenced by: (None)
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