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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
StepHypRef 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
32eqeq1i 2090 . . 3  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
43exbii 1537 . 2  |-  ( E. x { x  |  x  e.  A }  =  { x }  <->  E. x  A  =  { x } )
51, 4bitri 182 1  |-  ( E! x  x  e.  A  <->  E. x  A  =  {
x } )
Colors of variables: wff set class
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)
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