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Theorem indistps2 21612
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 21611. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 21613 and indistps2ALT 21614 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 2908 . . . 4 𝐴 ∈ V
63, 5unipr 4843 . . 3 {∅, 𝐴} = (∅ ∪ 𝐴)
7 uncom 4127 . . 3 (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8 un0 4342 . . 3 (𝐴 ∪ ∅) = 𝐴
96, 7, 83eqtrri 2847 . 2 𝐴 = {∅, 𝐴}
10 indistop 21602 . 2 {∅, 𝐴} ∈ Top
111, 2, 9, 10istpsi 21542 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  Vcvv 3493  cun 3932  c0 4289  {cpr 4561   cuni 4830  cfv 6348  Basecbs 16475  TopOpenctopn 16687  TopSpctps 21532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 21494  df-topon 21511  df-topsp 21533
This theorem is referenced by: (None)
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