Theorem elsn2 3652

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 3651	. 2 |- ( B e. _V -> ( A e. { B } <-> A = B ) )
3	1, 2	ax-mp 5	1 |- ( A e. { B } <-> A = B )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   {csn 3618

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sn 3624

This theorem is referenced by:  el1o  6490  elnn0  9242  elxnn0  9305  fisumss  11535  fprodssdc  11733  rest0  14347