Theorem elsn2 3707

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 3706	. 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 1398   e. wcel 2202   _Vcvv 2803   {csn 3673

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sn 3679

This theorem is referenced by:  el1o  6648  elnn0  9463  elxnn0  9528  fisumss  12033  fprodssdc  12231  rest0  14990