Theorem indistps2ALT 20628

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 20626 from the structural version indistps 20625. (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 6113	. . . 4 ⊢ (Base‘𝐾) ∈ V
3	1, 2	eqeltrri 2685	. . 3 ⊢ 𝐴 ∈ V
4		indistopon 20615	. . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
5	3, 4	ax-mp 5	. 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴)
6	1	eqcomi 2619	. . 3 ⊢ 𝐴 = (Base‘𝐾)
7		indistps2ALT.j	. . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴}
8	7	eqcomi 2619	. . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾)
9	6, 8	istps 20551	. 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴))
10	5, 9	mpbir 220	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {cpr 4127  ‘cfv 5804  Basecbs 15695  TopOpenctopn 15905  TopOnctopon 20518  TopSpctps 20519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523  df-topsp 20524

This theorem is referenced by: (None)