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Theorem indistop 20616
 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 20614 . 2 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2 fvex 6113 . . . 4 ( I ‘𝐴) ∈ V
3 indistopon 20615 . . . 4 (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
42, 3ax-mp 5 . . 3 {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
54topontopi 20546 . 2 {∅, ( I ‘𝐴)} ∈ Top
61, 5eqeltrri 2685 1 {∅, 𝐴} ∈ Top
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {cpr 4127   I cid 4948  ‘cfv 5804  Topctop 20517  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523 This theorem is referenced by:  indistpsx  20624  indistps  20625  indistps2  20626  indiscld  20705  indiscon  21031  txindis  21247  indispcon  30470  onpsstopbas  31599
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