Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  snidb Structured version   Visualization version   GIF version

Theorem snidb 4353
 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
StepHypRef Expression
1 snidg 4352 . 2 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2 elex 3353 . 2 (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
31, 2impbii 199 1 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∈ wcel 2140  Vcvv 3341  {csn 4322 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-sn 4323 This theorem is referenced by:  snid  4354  dffv2  6435
 Copyright terms: Public domain W3C validator