Theorem elsncg 3983

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
elsncg	|- ( A e. V -> ( A e. { B } <-> A = B ) )

Proof of Theorem elsncg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2475	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3960	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 3175	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1452   e. wcel 1904   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-sn 3960

This theorem is referenced by:  elsnc  3984  elsni  3985  snidg  3986  nelsn  3992  eltpg  4005  eldifsn  4088  elsucg  5497  ltxr  11438  elfzp12  11899  fprodn0f  14122  lcmfunsnlem2  14692  ramcl  15066  initoeu2lem1  15987  pmtrdifellem4  17198  logbmpt  23804  nbcusgra  25270  frgrancvvdeqlem1  25837  xrge0tsmsbi  28623  elzrhunit  28857  elzdif0  28858  esumrnmpt2  28963  plymulx  29509  bj-projval  31660  unirnmapsn  37567  reclimc  37831  itgsincmulx  37948  dirkercncflem2  38078  dirkercncflem4  38080  fourierdlem53  38135  fourierdlem58  38140  fourierdlem60  38142  fourierdlem61  38143  fourierdlem62  38144  fourierdlem76  38158  fourierdlem101  38183  elaa2  38211  etransc  38261  qndenserrnbl  38276  sge0tsms  38336  sge0fodjrnlem  38372  el1fzopredsuc  38867