Theorem snidb 3667

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

Step	Hyp	Ref	Expression
1		snidg 3666	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2		elex 2785	. 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
3	1, 2	impbii 126	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})

Syntax hints:   ↔ wb 105   ∈ wcel 2177  Vcvv 2773  {csn 3637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sn 3643

This theorem is referenced by:  snid  3668