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Theorem eusn 3605
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 3601 . 2  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
2 abid2 2261 . . . 4  |-  { x  |  x  e.  A }  =  A
32eqeq1i 2148 . . 3  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
43exbii 1585 . 2  |-  ( E. x { x  |  x  e.  A }  =  { x }  <->  E. x  A  =  { x } )
51, 4bitri 183 1  |-  ( E! x  x  e.  A  <->  E. x  A  =  {
x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1332   E.wex 1469    e. wcel 1481   E!weu 2000   {cab 2126   {csn 3532
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3538
This theorem is referenced by: (None)
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