Theorem snidb 4596

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 4595	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2		elex 3450	. 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
3	1, 2	impbii 208	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})

Syntax hints:   ↔ wb 205   ∈ wcel 2106  Vcvv 3432  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-sn 4562

This theorem is referenced by:  snid  4597  dffv2  6863  snen1el  41132