Theorem indistps2 23074

Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 23073. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 23075 and indistps2ALT 23076 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

Step	Hyp	Ref	Expression
1		indistps2.a	. 2 ⊢ (Base‘𝐾) = 𝐴
2		indistps2.j	. 2 ⊢ (TopOpen‘𝐾) = {∅, 𝐴}
3		0ex 5259	. . . 4 ⊢ ∅ ∈ V
4		fvex 6882	. . . . 5 ⊢ (Base‘𝐾) ∈ V
5	1, 4	eqeltrri 2861	. . . 4 ⊢ 𝐴 ∈ V
6	3, 5	unipr 4884	. . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴)
7		uncom 4113	. . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8		un0 4350	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
9	6, 7, 8	3eqtrri 2792	. 2 ⊢ 𝐴 = ∪ {∅, 𝐴}
10		indistop 23064	. 2 ⊢ {∅, 𝐴} ∈ Top
11	1, 2, 9, 10	istpsi 23004	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ∪ cun 3904  ∅c0 4287  {cpr 4586  ∪ cuni 4867  ‘cfv 6523  Basecbs 17247  TopOpenctopn 17452  TopSpctps 22994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-top 22956  df-topon 22973  df-topsp 22995

This theorem is referenced by: (None)