Theorem elsng 3659

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 2214	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3650	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2928	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   {csn 3644

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2779  df-sn 3650

This theorem is referenced by:  elsn  3660  elsni  3662  snidg  3673  eltpg  3689  eldifsn  3772  elsucg  4470  funconstss  5723  fniniseg  5725  fniniseg2  5727  tpfidceq  7055  fidcenumlemrks  7083  ltxr  9934  elfzp12  10258  1exp  10752  imasaddfnlemg  13307  0subm  13477  0subg  13696  0nsg  13711  kerf1ghm  13771  lsssn0  14293  plycj  15394  2lgslem2  15730