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Theorem indistps2 22378
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 22377. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22379 and indistps2ALT 22381 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)
Hypotheses
Ref Expression
indistps2.a (Baseβ€˜πΎ) = 𝐴
indistps2.j (TopOpenβ€˜πΎ) = {βˆ…, 𝐴}
Assertion
Ref Expression
indistps2 𝐾 ∈ TopSp

Proof of Theorem indistps2
StepHypRef Expression
1 indistps2.a . 2 (Baseβ€˜πΎ) = 𝐴
2 indistps2.j . 2 (TopOpenβ€˜πΎ) = {βˆ…, 𝐴}
3 0ex 5265 . . . 4 βˆ… ∈ V
4 fvex 6856 . . . . 5 (Baseβ€˜πΎ) ∈ V
51, 4eqeltrri 2831 . . . 4 𝐴 ∈ V
63, 5unipr 4884 . . 3 βˆͺ {βˆ…, 𝐴} = (βˆ… βˆͺ 𝐴)
7 uncom 4114 . . 3 (βˆ… βˆͺ 𝐴) = (𝐴 βˆͺ βˆ…)
8 un0 4351 . . 3 (𝐴 βˆͺ βˆ…) = 𝐴
96, 7, 83eqtrri 2766 . 2 𝐴 = βˆͺ {βˆ…, 𝐴}
10 indistop 22368 . 2 {βˆ…, 𝐴} ∈ Top
111, 2, 9, 10istpsi 22307 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107  Vcvv 3444   βˆͺ cun 3909  βˆ…c0 4283  {cpr 4589  βˆͺ cuni 4866  β€˜cfv 6497  Basecbs 17088  TopOpenctopn 17308  TopSpctps 22297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-top 22259  df-topon 22276  df-topsp 22298
This theorem is referenced by: (None)
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