Theorem snidb 4669

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 4668	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2		elex 3502	. 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
3	1, 2	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3481  {csn 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3483  df-sn 4635

This theorem is referenced by:  snid  4670  dffv2  7011  snen1el  43531