Theorem indistps2 22954

Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 22953. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22955 and indistps2ALT 22956 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 5250	. . . 4 ⊢ ∅ ∈ V
4		fvex 6845	. . . . 5 ⊢ (Base‘𝐾) ∈ V
5	1, 4	eqeltrri 2831	. . . 4 ⊢ 𝐴 ∈ V
6	3, 5	unipr 4878	. . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴)
7		uncom 4108	. . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8		un0 4344	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
9	6, 7, 8	3eqtrri 2762	. 2 ⊢ 𝐴 = ∪ {∅, 𝐴}
10		indistop 22944	. 2 ⊢ {∅, 𝐴} ∈ Top
11	1, 2, 9, 10	istpsi 22884	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∪ cun 3897  ∅c0 4283  {cpr 4580  ∪ cuni 4861  ‘cfv 6490  Basecbs 17134  TopOpenctopn 17339  TopSpctps 22874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-top 22836  df-topon 22853  df-topsp 22875

This theorem is referenced by: (None)