Theorem elsn2g 3676

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 3661	. 2 |- ( A e. { B } -> A = B )
2		snidg 3672	. . 3 |- ( B e. V -> B e. { B } )
3		eleq1 2270	. . 3 |- ( A = B -> ( A e. { B } <-> B e. { B } ) )
4	2, 3	syl5ibrcom 157	. 2 |- ( B e. V -> ( A = B -> A e. { B } ) )
5	1, 4	impbid2 143	1 |- ( B e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   {csn 3643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sn 3649

This theorem is referenced by:  elsn2  3677  elsuc2g  4470  mptiniseg  5196  elfzp1  10229  fzosplitsni  10401  zfz1isolemiso  11021  1nsgtrivd  13670  zrhrhmb  14499  ply1termlem  15329