Theorem indistps2ALT 22934

Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 22931 from the structural version indistps 22930. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
indistps2ALT.a	⊢ (Base‘𝐾) = 𝐴
indistps2ALT.j	⊢ (TopOpen‘𝐾) = {∅, 𝐴}

Assertion

Ref	Expression
indistps2ALT	⊢ 𝐾 ∈ TopSp

Proof of Theorem indistps2ALT

Step	Hyp	Ref	Expression
1		indistps2ALT.a	. . . 4 ⊢ (Base‘𝐾) = 𝐴
2		fvex 6904	. . . 4 ⊢ (Base‘𝐾) ∈ V
3	1, 2	eqeltrri 2822	. . 3 ⊢ 𝐴 ∈ V
4		indistopon 22920	. . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
5	3, 4	ax-mp 5	. 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴)
6	1	eqcomi 2734	. . 3 ⊢ 𝐴 = (Base‘𝐾)
7		indistps2ALT.j	. . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴}
8	7	eqcomi 2734	. . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾)
9	6, 8	istps 22852	. 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴))
10	5, 9	mpbir 230	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3463  ∅c0 4318  {cpr 4626  ‘cfv 6542  Basecbs 17177  TopOpenctopn 17400  TopOnctopon 22828  TopSpctps 22850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-top 22812  df-topon 22829  df-topsp 22851

This theorem is referenced by: (None)