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Theorem indisuni 20930
 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 20927 . . 3 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2 fvex 6314 . . . 4 ( I ‘𝐴) ∈ V
3 indistopon 20928 . . . 4 (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
42, 3ax-mp 5 . . 3 {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
51, 4eqeltrri 2800 . 2 {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴))
65toponunii 20844 1 ( I ‘𝐴) = {∅, 𝐴}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1596   ∈ wcel 2103  Vcvv 3304  ∅c0 4023  {cpr 4287  ∪ cuni 4544   I cid 5127  ‘cfv 6001  TopOnctopon 20838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-top 20822  df-topon 20839 This theorem is referenced by:  indiscld  21018  indisconn  21344  txindis  21560
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