Theorem elsn 3725

Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Hypothesis

Ref	Expression
elsn.1	|- A e. _V

Assertion

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

Proof of Theorem elsn

Step	Hyp	Ref	Expression
1		elsn.1	. 2 |- A e. _V
2		elsng 3724	. 2 |- ( A 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:  velsn  3726  sneqr  3885  onsucelsucexmid  4677  ordsoexmid  4709  opthprc  4826  dmsnm  5253  dmsnopg  5259  cnvcnvsn  5264  sniota  5368  fsn  5880  eusvobj2  6071  mapdm0  6937  djulclb  7395  pw1nel3  7590  sucpw1nel3  7592  opelreal  8194  hashf1lem2  11286