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Theorem snidb 4664
Description: A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
snidb (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})

Proof of Theorem snidb
StepHypRef Expression
1 snidg 4663 . 2 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2 elex 3490 . 2 (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
31, 2impbii 208 1 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2099  Vcvv 3471  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-sn 4630
This theorem is referenced by:  snid  4665  dffv2  6993  snen1el  42955
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