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Theorem snidb 3525
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 3524 . 2 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2 elex 2671 . 2 (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
31, 2impbii 125 1 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1465  Vcvv 2660  {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sn 3503
This theorem is referenced by:  snid  3526
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