Theorem elsng 3591

Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

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

Proof of Theorem elsng

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2172	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3582	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2873	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sn 3582

This theorem is referenced by:  elsn  3592  elsni  3594  snidg  3605  eltpg  3621  eldifsn  3703  elsucg  4382  funconstss  5603  fniniseg  5605  fniniseg2  5607  fidcenumlemrks  6918  ltxr  9711  elfzp12  10034  1exp  10484