Theorem snidg 2437

Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.

Assertion

Ref	Expression
snidg	|- (A e. B -> A e. {A})

Proof of Theorem snidg

Step	Hyp	Ref	Expression
1		eqid 1478	. 2 |- A = A
2		elsncg 2434	. 2 |- (A e. B -> (A e. {A} <-> A = A))
3	1, 2	mpbiri 194	1 |- (A e. B -> A e. {A})

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  {csn 2413

This theorem is referenced by:  snidb 2438  elsnc2g 2440  disjsn 2445  curry1 4104  supsnALT 4601  cfsuc 4927  infpss 7575  oefil2 10552  cnfilca 10562

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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