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Theorem snidb 4346
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 4345 . 2 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2 elex 3364 . 2 (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
31, 2impbii 199 1 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2145  Vcvv 3351  {csn 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sn 4317
This theorem is referenced by:  snid  4347  dffv2  6413
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