Theorem indistps2 22977

Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 22976. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22978 and indistps2ALT 22979 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 5242	. . . 4 ⊢ ∅ ∈ V
4		fvex 6853	. . . . 5 ⊢ (Base‘𝐾) ∈ V
5	1, 4	eqeltrri 2833	. . . 4 ⊢ 𝐴 ∈ V
6	3, 5	unipr 4867	. . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴)
7		uncom 4098	. . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8		un0 4334	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
9	6, 7, 8	3eqtrri 2764	. 2 ⊢ 𝐴 = ∪ {∅, 𝐴}
10		indistop 22967	. 2 ⊢ {∅, 𝐴} ∈ Top
11	1, 2, 9, 10	istpsi 22907	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887  ∅c0 4273  {cpr 4569  ∪ cuni 4850  ‘cfv 6498  Basecbs 17179  TopOpenctopn 17384  TopSpctps 22897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-top 22859  df-topon 22876  df-topsp 22898

This theorem is referenced by: (None)