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Theorem indislem 22931
Description: A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
indislem {∅, ( I ‘𝐴)} = {∅, 𝐴}

Proof of Theorem indislem
StepHypRef Expression
1 fvi 6979 . . 3 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
21preq2d 4749 . 2 (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3 dfsn2 4645 . . . 4 {∅} = {∅, ∅}
43eqcomi 2737 . . 3 {∅, ∅} = {∅}
5 fvprc 6894 . . . 4 𝐴 ∈ V → ( I ‘𝐴) = ∅)
65preq2d 4749 . . 3 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7 prprc2 4775 . . 3 𝐴 ∈ V → {∅, 𝐴} = {∅})
84, 6, 73eqtr4a 2794 . 2 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
92, 8pm2.61i 182 1 {∅, ( I ‘𝐴)} = {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3473  c0 4326  {csn 4632  {cpr 4634   I cid 5579  cfv 6553
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-pr 5433
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-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-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  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
This theorem is referenced by:  indistop  22933  indisuni  22934  indiscld  23023  indisconn  23350  txindis  23566  hmphindis  23729
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