Theorem eusn 4265

Description: Two ways to express " A is a singleton." (Contributed by NM, 30-Oct-2010.)

Assertion

Ref	Expression
eusn	|- ( E! x x e. A <-> E. x A = { x } )

Distinct variable group:   x, A

Proof of Theorem eusn

Step	Hyp	Ref	Expression
1		euabsn 4261	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2745	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2627	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1774	. 2 |- ( E. x { x | x e. A } = { x } <-> E. x A = { x } )
5	1, 4	bitri 264	1 |- ( E! x x e. A <-> E. x A = { x } )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   {cab 2608   {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-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178

This theorem is referenced by:  initoid  16655  termoid  16656  initoeu2lem1  16664  funpartfv  32052  irinitoringc  42069