indistps2 - Metamath Proof Explorer

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Theorem indistps2 22908

Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 22907. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22909 and indistps2ALT 22911 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)

**Hypotheses**
Ref	Expression
indistps2.a	⊢ (Base‘𝐾) = 𝐴
indistps2.j	⊢ (TopOpen‘𝐾) = {∅, 𝐴}

**Assertion**
Ref	Expression
indistps2	⊢ 𝐾 ∈ TopSp

**Proof of Theorem indistps2**
Step	Hyp	Ref	Expression
1		indistps2.a	. 2 ⊢ (Base‘𝐾) = 𝐴
2		indistps2.j	. 2 ⊢ (TopOpen‘𝐾) = {∅, 𝐴}
3		0ex 5301	. . . 4 ⊢ ∅ ∈ V
4		fvex 6904	. . . . 5 ⊢ (Base‘𝐾) ∈ V
5	1, 4	eqeltrri 2826	. . . 4 ⊢ 𝐴 ∈ V
6	3, 5	unipr 4920	. . 3 ⊢ ∪ {∅, 𝐴} = (∅ ∪ 𝐴)
7		uncom 4149	. . 3 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
8		un0 4386	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
9	6, 7, 8	3eqtrri 2761	. 2 ⊢ 𝐴 = ∪ {∅, 𝐴}
10		indistop 22898	. 2 ⊢ {∅, 𝐴} ∈ Top
11	1, 2, 9, 10	istpsi 22837	1 ⊢ 𝐾 ∈ TopSp

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3470 ∪ cun 3943 ∅c0 4318 {cpr 4626 ∪ cuni 4903 ‘cfv 6542 Basecbs 17173 TopOpenctopn 17396 TopSpctps 22827

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 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 22789 df-topon 22806 df-topsp 22828

This theorem is referenced by: (None)

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