Theorem elsng 3596

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 2177	. 2 |- ( x = A -> ( x = B <-> A = B ) )
2		df-sn 3587	. 2 |- { B } = { x | x = B }
3	1, 2	elab2g 2877	1 |- ( A e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   {csn 3581

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3587

This theorem is referenced by:  elsn  3597  elsni  3599  snidg  3610  eltpg  3626  eldifsn  3708  elsucg  4387  funconstss  5612  fniniseg  5614  fniniseg2  5616  fidcenumlemrks  6927  ltxr  9721  elfzp12  10044  1exp  10494  0subm  12691