Theorem indistop 22504

Description: The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
indistop	⊢ {∅, 𝐴} ∈ Top

Proof of Theorem indistop

Step	Hyp	Ref	Expression
1		indislem 22502	. 2 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
2		fvex 6904	. . . 4 ⊢ ( I ‘𝐴) ∈ V
3		indistopon 22503	. . . 4 ⊢ (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
4	2, 3	ax-mp 5	. . 3 ⊢ {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
5	4	topontopi 22416	. 2 ⊢ {∅, ( I ‘𝐴)} ∈ Top
6	1, 5	eqeltrri 2830	1 ⊢ {∅, 𝐴} ∈ Top

Syntax hints:   ∈ wcel 2106  Vcvv 3474  ∅c0 4322  {cpr 4630   I cid 5573  ‘cfv 6543  Topctop 22394  TopOnctopon 22411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22395  df-topon 22412

This theorem is referenced by:  indistpsx  22512  indistps  22513  indistps2  22514  indiscld  22594  indisconn  22921  txindis  23137  indispconn  34220  onpsstopbas  35310