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Theorem eusn 3536
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 3532 . 2  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
2 abid2 2215 . . . 4  |-  { x  |  x  e.  A }  =  A
32eqeq1i 2102 . . 3  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
43exbii 1548 . 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 1296   E.wex 1433    e. wcel 1445   E!weu 1955   {cab 2081   {csn 3466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-sn 3472
This theorem is referenced by: (None)
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