Theorem elsng 3622

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 2196	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3613	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2899	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2160   {csn 3607

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sn 3613

This theorem is referenced by:  elsn  3623  elsni  3625  snidg  3636  eltpg  3652  eldifsn  3734  elsucg  4422  funconstss  5655  fniniseg  5657  fniniseg2  5659  fidcenumlemrks  6983  ltxr  9807  elfzp12  10131  1exp  10583  imasaddfnlemg  12794  0subm  12951  0subg  13155  0nsg  13170  kerf1ghm  13230  lsssn0  13703