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Theorem indistop 21609
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 21607 . 2 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2 fvex 6682 . . . 4 ( I ‘𝐴) ∈ V
3 indistopon 21608 . . . 4 (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
42, 3ax-mp 5 . . 3 {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
54topontopi 21522 . 2 {∅, ( I ‘𝐴)} ∈ Top
61, 5eqeltrri 2910 1 {∅, 𝐴} ∈ Top
Colors of variables: wff setvar class
Syntax hints:  wcel 2110  Vcvv 3494  c0 4290  {cpr 4568   I cid 5458  cfv 6354  Topctop 21500  TopOnctopon 21517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-top 21501  df-topon 21518
This theorem is referenced by:  indistpsx  21617  indistps  21618  indistps2  21619  indiscld  21698  indisconn  22025  txindis  22241  indispconn  32481  onpsstopbas  33778
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