MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indistop Structured version   Visualization version   GIF version

Theorem indistop 21538
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 21536 . 2 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2 fvex 6676 . . . 4 ( I ‘𝐴) ∈ V
3 indistopon 21537 . . . 4 (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
42, 3ax-mp 5 . . 3 {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
54topontopi 21451 . 2 {∅, ( I ‘𝐴)} ∈ Top
61, 5eqeltrri 2907 1 {∅, 𝐴} ∈ Top
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  Vcvv 3492  c0 4288  {cpr 4559   I cid 5452  cfv 6348  Topctop 21429  TopOnctopon 21446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-topon 21447
This theorem is referenced by:  indistpsx  21546  indistps  21547  indistps2  21548  indiscld  21627  indisconn  21954  txindis  22170  indispconn  32378  onpsstopbas  33675
  Copyright terms: Public domain W3C validator