Theorem elsn2g 3616

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 28-Oct-2003.)

Assertion

Ref	Expression
elsn2g	|- ( B e. V -> ( A e. { B } <-> A = B ) )

Proof of Theorem elsn2g

Step	Hyp	Ref	Expression
1		elsni 3601	. 2 |- ( A e. { B } -> A = B )
2		snidg 3612	. . 3 |- ( B e. V -> B e. { B } )
3		eleq1 2233	. . 3 |- ( A = B -> ( A e. { B } <-> B e. { B } ) )
4	2, 3	syl5ibrcom 156	. 2 |- ( B e. V -> ( A = B -> A e. { B } ) )
5	1, 4	impbid2 142	1 |- ( B e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   {csn 3583

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 3589

This theorem is referenced by:  elsn2  3617  elsuc2g  4390  mptiniseg  5105  elfzp1  10028  fzosplitsni  10191  zfz1isolemiso  10774