Theorem snidb 2438

Description: A class is a set iff it is a member of its singleton.

Assertion

Ref	Expression
snidb	|- (A e. V <-> A e. {A})

Proof of Theorem snidb

Step	Hyp	Ref	Expression
1		snidg 2437	. 2 |- (A e. V -> A e. {A})
2		elisset 1820	. 2 |- (A e. {A} -> A e. V)
3	1, 2	impbi 157	1 |- (A e. V <-> A e. {A})

Syntax hints:   <-> wb 146   e. wcel 960  Vcvv 1814  {csn 2413

This theorem is referenced by:  snid 2439

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417

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