Theorem elsng 3681

Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

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

Proof of Theorem elsng

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2236	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3672	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2950	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sn 3672

This theorem is referenced by:  elsn  3682  elsni  3684  snidg  3695  eltpg  3711  eldifsn  3795  elsucg  4495  funconstss  5753  fniniseg  5755  fniniseg2  5757  tpfidceq  7092  fidcenumlemrks  7120  ltxr  9971  elfzp12  10295  1exp  10790  imasaddfnlemg  13347  0subm  13517  0subg  13736  0nsg  13751  kerf1ghm  13811  lsssn0  14334  plycj  15435  2lgslem2  15771