Theorem snidb 4626

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 4625	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2		elex 3475	. 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
3	1, 2	impbii 212	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2142  Vcvv 3454  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-sn 4589

This theorem is used by:  snid  4627  dffv2  6976  snen1el  44279