Theorem elsn2g 3404

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 3393	. 2 |- ( A e. { B } -> A = B )
2		snidg 3400	. . 3 |- ( B e. V -> B e. { B } )
3		eleq1 2100	. . 3 |- ( A = B -> ( A e. { B } <-> B e. { B } ) )
4	2, 3	syl5ibrcom 146	. 2 |- ( B e. V -> ( A = B -> A e. { B } ) )
5	1, 4	impbid2 131	1 |- ( B e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by:  elsn2  3405  elsuc2g  4142  mptiniseg  4815  elfzp1  8934  fzosplitsni  9091  iseqid3  9245