Theorem elsn 3649

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 3648	. 2 |- ( A e. _V -> ( A e. { B } <-> A = B ) )
3	1, 2	ax-mp 5	1 |- ( A e. { B } <-> A = B )

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2176   _Vcvv 2772   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sn 3639

This theorem is referenced by:  velsn  3650  sneqr  3801  onsucelsucexmid  4579  ordsoexmid  4611  opthprc  4727  dmsnm  5149  dmsnopg  5155  cnvcnvsn  5160  sniota  5263  fsn  5754  eusvobj2  5932  mapdm0  6752  djulclb  7159  pw1nel3  7345  sucpw1nel3  7347  opelreal  7942