ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snidb GIF version

Theorem snidb 3637
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 3636 . 2 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
2 elex 2763 . 2 (𝐴 ∈ {𝐴} → 𝐴 ∈ V)
31, 2impbii 126 1 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2160  Vcvv 2752  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sn 3613
This theorem is referenced by:  snid  3638
  Copyright terms: Public domain W3C validator