Theorem eusn 3485

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 3481	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2203	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2090	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1537	. 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 1285   E.wex 1422   e. wcel 1434   E!weu 1943   {cab 2069   {csn 3417

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sn 3423

This theorem is referenced by: (None)