Theorem eusn 3650

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 3646	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2287	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2173	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1593	. 2 |- ( E. x { x | x e. A } = { x } <-> E. x A = { x } )
5	1, 4	bitri 183	1 |- ( E! x x e. A <-> E. x A = { x } )

Syntax hints:   <-> wb 104   = wceq 1343   E.wex 1480   E!weu 2014   e. wcel 2136   {cab 2151   {csn 3576

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sn 3582

This theorem is referenced by: (None)