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Theorem snidb 3562
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 3561 . 2 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2 elex 2700 . 2 (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
31, 2impbii 125 1 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1481  Vcvv 2689  {csn 3532
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3538
This theorem is referenced by:  snid  3563
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