Theorem elsn2g 4210

Description: There is exactly 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 4194	. 2 |- ( A e. { B } -> A = B )
2		snidg 4206	. . 3 |- ( B e. V -> B e. { B } )
3		eleq1 2689	. . 3 |- ( A = B -> ( A e. { B } <-> B e. { B } ) )
4	2, 3	syl5ibrcom 237	. 2 |- ( B e. V -> ( A = B -> A e. { B } ) )
5	1, 4	impbid2 216	1 |- ( B e. V -> ( A e. { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178

This theorem is referenced by:  elsn2  4211  mptiniseg  5629  elsuc2g  5793  extmptsuppeq  7319  fzosplitsni  12579  limcco  23657  ply1termlem  23959  elpmapat  35050  stirlinglem8  40298  dirkercncflem2  40321