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Theorem indistop 21177
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 21175 . 2 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2 fvex 6446 . . . 4 ( I ‘𝐴) ∈ V
3 indistopon 21176 . . . 4 (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
42, 3ax-mp 5 . . 3 {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
54topontopi 21090 . 2 {∅, ( I ‘𝐴)} ∈ Top
61, 5eqeltrri 2903 1 {∅, 𝐴} ∈ Top
Colors of variables: wff setvar class
Syntax hints:  wcel 2164  Vcvv 3414  c0 4144  {cpr 4399   I cid 5249  cfv 6123  Topctop 21068  TopOnctopon 21085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-top 21069  df-topon 21086
This theorem is referenced by:  indistpsx  21185  indistps  21186  indistps2  21187  indiscld  21266  indisconn  21592  txindis  21808  indispconn  31751  onpsstopbas  32951
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