Theorem indistps2 22956

Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 22955. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22957 and indistps2ALT 22958 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 5252	. . . 4 ⊢ ∅ ∈ V
4		fvex 6847	. . . . 5 ⊢ (Base‘𝐾) ∈ V
5	1, 4	eqeltrri 2833	. . . 4 ⊢ 𝐴 ∈ V
6	3, 5	unipr 4880	. . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴)
7		uncom 4110	. . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8		un0 4346	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
9	6, 7, 8	3eqtrri 2764	. 2 ⊢ 𝐴 = ∪ {∅, 𝐴}
10		indistop 22946	. 2 ⊢ {∅, 𝐴} ∈ Top
11	1, 2, 9, 10	istpsi 22886	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899  ∅c0 4285  {cpr 4582  ∪ cuni 4863  ‘cfv 6492  Basecbs 17136  TopOpenctopn 17341  TopSpctps 22876

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-top 22838  df-topon 22855  df-topsp 22877

This theorem is referenced by: (None)