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

Theorem indisuni 23026
Description: The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
indisuni ( I ‘𝐴) = {∅, 𝐴}

Proof of Theorem indisuni
StepHypRef Expression
1 indislem 23023 . . 3 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2 fvex 6920 . . . 4 ( I ‘𝐴) ∈ V
3 indistopon 23024 . . . 4 (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
42, 3ax-mp 5 . . 3 {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
51, 4eqeltrri 2836 . 2 {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴))
65toponunii 22938 1 ( I ‘𝐴) = {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {cpr 4633   cuni 4912   I cid 5582  cfv 6563  TopOnctopon 22932
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-top 22916  df-topon 22933
This theorem is referenced by:  indiscld  23115  indisconn  23442  txindis  23658
  Copyright terms: Public domain W3C validator