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Theorem indisuni 22926
Description: The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
indisuni ( I β€˜π΄) = βˆͺ {βˆ…, 𝐴}

Proof of Theorem indisuni
StepHypRef Expression
1 indislem 22923 . . 3 {βˆ…, ( I β€˜π΄)} = {βˆ…, 𝐴}
2 fvex 6915 . . . 4 ( I β€˜π΄) ∈ V
3 indistopon 22924 . . . 4 (( I β€˜π΄) ∈ V β†’ {βˆ…, ( I β€˜π΄)} ∈ (TopOnβ€˜( I β€˜π΄)))
42, 3ax-mp 5 . . 3 {βˆ…, ( I β€˜π΄)} ∈ (TopOnβ€˜( I β€˜π΄))
51, 4eqeltrri 2826 . 2 {βˆ…, 𝐴} ∈ (TopOnβ€˜( I β€˜π΄))
65toponunii 22838 1 ( I β€˜π΄) = βˆͺ {βˆ…, 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3473  βˆ…c0 4326  {cpr 4634  βˆͺ cuni 4912   I cid 5579  β€˜cfv 6553  TopOnctopon 22832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-top 22816  df-topon 22833
This theorem is referenced by:  indiscld  23015  indisconn  23342  txindis  23558
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