Theorem indistps2 20816

Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 20815. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 20817 and indistps2ALT 20818 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 4790	. . . 4 ⊢ ∅ ∈ V
4		fvex 6201	. . . . 5 ⊢ (Base‘𝐾) ∈ V
5	1, 4	eqeltrri 2698	. . . 4 ⊢ 𝐴 ∈ V
6	3, 5	unipr 4449	. . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴)
7		uncom 3757	. . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8		un0 3967	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
9	6, 7, 8	3eqtrri 2649	. 2 ⊢ 𝐴 = ∪ {∅, 𝐴}
10		indistop 20806	. 2 ⊢ {∅, 𝐴} ∈ Top
11	1, 2, 9, 10	istpsi 20746	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {cpr 4179  ∪ cuni 4436  ‘cfv 5888  Basecbs 15857  TopOpenctopn 16082  TopSpctps 20736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-topon 20716  df-topsp 20737

This theorem is referenced by: (None)