Theorem elsn2 3743

Description: There is only 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 3742	. 2 |- ( B e. _V -> ( A e. { B } <-> A = B ) )
3	1, 2	ax-mp 5	1 |- ( A e. { B } <-> A = B )

This proof depends on syntax axioms:   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3709

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3715

This theorem is used by:  el1o  6710  indconst0  9302  indconst1  9303  elnn0  9565  elxnn0  9632  fisumss  12159  fprodssdc  12357  ballotfilemcdc  13223  rest0  15280