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Theorem snidb 3478
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 3477 . 2 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2 elex 2631 . 2 (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
31, 2impbii 125 1 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1439  Vcvv 2620  {csn 3450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-sn 3456
This theorem is referenced by:  snid  3479
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