Theorem indistps2ALT 22518

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 22515 from the structural version indistps 22514. (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 6905	. . . 4 ⊢ (Base‘𝐾) ∈ V
3	1, 2	eqeltrri 2831	. . 3 ⊢ 𝐴 ∈ V
4		indistopon 22504	. . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
5	3, 4	ax-mp 5	. 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴)
6	1	eqcomi 2742	. . 3 ⊢ 𝐴 = (Base‘𝐾)
7		indistps2ALT.j	. . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴}
8	7	eqcomi 2742	. . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾)
9	6, 8	istps 22436	. 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴))
10	5, 9	mpbir 230	1 ⊢ 𝐾 ∈ TopSp

Syntax hints:   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ∅c0 4323  {cpr 4631  ‘cfv 6544  Basecbs 17144  TopOpenctopn 17367  TopOnctopon 22412  TopSpctps 22434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-top 22396  df-topon 22413  df-topsp 22435

This theorem is referenced by: (None)