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Theorem indistop 21006
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
StepHypRef Expression
1 indislem 21004 . 2 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2 fvex 6360 . . . 4 ( I ‘𝐴) ∈ V
3 indistopon 21005 . . . 4 (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
42, 3ax-mp 5 . . 3 {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
54topontopi 20920 . 2 {∅, ( I ‘𝐴)} ∈ Top
61, 5eqeltrri 2834 1 {∅, 𝐴} ∈ Top
Colors of variables: wff setvar class
Syntax hints:  wcel 2137  Vcvv 3338  c0 4056  {cpr 4321   I cid 5171  cfv 6047  Topctop 20898  TopOnctopon 20915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-iota 6010  df-fun 6049  df-fv 6055  df-top 20899  df-topon 20916
This theorem is referenced by:  indistpsx  21014  indistps  21015  indistps2  21016  indiscld  21095  indisconn  21421  txindis  21637  indispconn  31521  onpsstopbas  32733
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