Theorem eusn 3466

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 3462	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2199	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2088	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1536	. 2 |- ( E. x { x | x e. A } = { x } <-> E. x A = { x } )
5	1, 4	bitri 182	1 |- ( E! x x e. A <-> E. x A = { x } )

Syntax hints:   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   E!weu 1941   {cab 2067   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sn 3404

This theorem is referenced by: (None)