Theorem elsncg 4056

Description: There is only 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 2471	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 4034	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 3257	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1379   e. wcel 1767   {csn 4033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-sn 4034

This theorem is referenced by:  elsnc  4057  elsni  4058  snidg  4059  eltpg  4075  eldifsn  4158  elsucg  4951  ltxr  11336  elfzp12  11769  ramcl  14423  pmtrdifellem4  16377  nbcusgra  24286  frgrancvvdeqlem1  24854  xrge0tsmsbi  27601  elzrhunit  27785  elzdif0  27786  plymulx  28330  reclimc  31518  itgsincmulx  31615  dirkercncflem2  31727  dirkercncflem4  31729  fourierdlem53  31783  fourierdlem58  31788  fourierdlem60  31790  fourierdlem61  31791  fourierdlem62  31792  fourierdlem76  31806  fourierdlem101  31831  bj-projval  34036