Theorem elsn2 4211

Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 12-Jun-1994.)

Hypothesis

Ref	Expression
elsn2.1	|- B e. _V

Assertion

Ref	Expression
elsn2	|- ( A e. { B } <-> A = B )

Proof of Theorem elsn2

Step	Hyp	Ref	Expression
1		elsn2.1	. 2 |- B e. _V
2		elsn2g 4210	. 2 |- ( B e. _V -> ( A e. { B } <-> A = B ) )
3	1, 2	ax-mp 5	1 |- ( A e. { B } <-> A = B )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178

This theorem is referenced by:  fparlem1  7277  fparlem2  7278  el1o  7579  fin1a2lem11  9232  fin1a2lem12  9233  elnn0  11294  elxnn0  11365  elfzp1  12391  fsumss  14456  fprodss  14678  elhoma  16682  islpidl  19246  zrhrhmb  19859  rest0  20973  qustgphaus  21926  taylfval  24113  elch0  28111  atoml2i  29242  bj-eltag  32965  bj-rest10b  33042  dibopelvalN  36432  dibopelval2  36434  climrec  39835