Theorem indistps2ALT 23140

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 23138 from the structural version indistps 23137. (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 6895	. . . 4 ⊢ (Base‘𝐾) ∈ V
3	1, 2	eqeltrri 2866	. . 3 ⊢ 𝐴 ∈ V
4		indistopon 23127	. . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
5	3, 4	ax-mp 5	. 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴)
6	1	eqcomi 2778	. . 3 ⊢ 𝐴 = (Base‘𝐾)
7		indistps2ALT.j	. . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴}
8	7	eqcomi 2778	. . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾)
9	6, 8	istps 23060	. 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴))
10	5, 9	mpbir 234	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1567   ∈ wcel 2149  Vcvv 3461  ∅c0 4292  {cpr 4594  ‘cfv 6537  Basecbs 17269  TopOpenctopn 17474  TopOnctopon 23036  TopSpctps 23058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-top 23020  df-topon 23037  df-topsp 23059

This theorem is referenced by: (None)