Theorem elsn2g 3427

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 28-Oct-2003.)

Assertion

Ref	Expression
elsn2g	|- ( B e. V -> ( A e. { B } <-> A = B ) )

Proof of Theorem elsn2g

Step	Hyp	Ref	Expression
1		elsni 3416	. 2 |- ( A e. { B } -> A = B )
2		snidg 3423	. . 3 |- ( B e. V -> B e. { B } )
3		eleq1 2141	. . 3 |- ( A = B -> ( A e. { B } <-> B e. { B } ) )
4	2, 3	syl5ibrcom 155	. 2 |- ( B e. V -> ( A = B -> A e. { B } ) )
5	1, 4	impbid2 141	1 |- ( B e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sn 3404

This theorem is referenced by:  elsn2  3428  elsuc2g  4160  mptiniseg  4835  elfzp1  9089  fzosplitsni  9244  iseqid3  9465