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Theorem indisuni 22369
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 22366 . . 3 {βˆ…, ( I β€˜π΄)} = {βˆ…, 𝐴}
2 fvex 6856 . . . 4 ( I β€˜π΄) ∈ V
3 indistopon 22367 . . . 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 2831 . 2 {βˆ…, 𝐴} ∈ (TopOnβ€˜( I β€˜π΄))
65toponunii 22281 1 ( I β€˜π΄) = βˆͺ {βˆ…, 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107  Vcvv 3444  βˆ…c0 4283  {cpr 4589  βˆͺ cuni 4866   I cid 5531  β€˜cfv 6497  TopOnctopon 22275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-top 22259  df-topon 22276
This theorem is referenced by:  indiscld  22458  indisconn  22785  txindis  23001
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