Theorem indistps2ALT 23332

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 23330 from the structural version indistps 23329. (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 6898	. . . 4 ⊢ (Base‘𝐾) ∈ V
3	1, 2	eqeltrri 2857	. . 3 ⊢ 𝐴 ∈ V
4		indistopon 23319	. . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
5	3, 4	ax-mp 5	. 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴)
6	1	eqcomi 2769	. . 3 ⊢ 𝐴 = (Base‘𝐾)
7		indistps2ALT.j	. . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴}
8	7	eqcomi 2769	. . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾)
9	6, 8	istps 23252	. 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴))
10	5, 9	mpbir 234	1 ⊢ 𝐾 ∈ TopSp

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  {cpr 4585  ‘cfv 6538  Basecbs 17387  TopOpenctopn 17592  TopOnctopon 23228  TopSpctps 23250

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-top 23212  df-topon 23229  df-topsp 23251

This theorem is used by: (None)