Theorem elsng 3456

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 2094	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3447	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2760	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   e. wcel 1438   {csn 3441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sn 3447

This theorem is referenced by:  elsn  3457  elsni  3459  snidg  3468  eltpg  3483  eldifsn  3562  elsucg  4222  funconstss  5401  fniniseg  5403  fniniseg2  5405  fidcenumlemrks  6641  ltxr  9215  elfzp12  9480  1exp  9949