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Theorem snidb 3432
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 3431 . 2 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2 elex 2611 . 2 (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
31, 2impbii 124 1 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Colors of variables: wff set class
Syntax hints:  wb 103  wcel 1434  Vcvv 2602  {csn 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sn 3412
This theorem is referenced by:  snid  3433
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