Theorem elsng 3658

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 3649	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2927	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   {csn 3643

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 2778  df-sn 3649

This theorem is referenced by:  elsn  3659  elsni  3661  snidg  3672  eltpg  3688  eldifsn  3771  elsucg  4469  funconstss  5721  fniniseg  5723  fniniseg2  5725  tpfidceq  7053  fidcenumlemrks  7081  ltxr  9932  elfzp12  10256  1exp  10750  imasaddfnlemg  13261  0subm  13431  0subg  13650  0nsg  13665  kerf1ghm  13725  lsssn0  14247  plycj  15348  2lgslem2  15684