Theorem elsng 3547

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 2147	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3538	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2835	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3538

This theorem is referenced by:  elsn  3548  elsni  3550  snidg  3561  eltpg  3576  eldifsn  3658  elsucg  4334  funconstss  5546  fniniseg  5548  fniniseg2  5550  fidcenumlemrks  6849  ltxr  9592  elfzp12  9910  1exp  10353