Theorem snidb 4621

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 4620	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2		elex 3471	. 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
3	1, 2	impbii 212	1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  Vcvv 3450  {csn 4583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sn 4584

This theorem is used by:  snid  4622  dffv2  6968  snen1el  44469