Theorem snidg 2423

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 1468	. 2 |- A = A
2		elsncg 2420	. 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 953   e. wcel 955  {csn 2399

This theorem is referenced by:  snidb 2424  elsnc2g 2426  disjsn 2431  opth1gOLD 2788  curry1 4082  supsnALT 4564  cfsuc 4887  oefil2 10441  cnfilca 10451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403

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