Theorem elsng 3648

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 2212	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3639	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2920	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sn 3639

This theorem is referenced by:  elsn  3649  elsni  3651  snidg  3662  eltpg  3678  eldifsn  3760  elsucg  4452  funconstss  5700  fniniseg  5702  fniniseg2  5704  tpfidceq  7029  fidcenumlemrks  7057  ltxr  9899  elfzp12  10223  1exp  10715  imasaddfnlemg  13179  0subm  13349  0subg  13568  0nsg  13583  kerf1ghm  13643  lsssn0  14165  plycj  15266  2lgslem2  15602