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Theorem eusn 3444
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 3440 . 2  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
2 abid2 2158 . . . 4  |-  { x  |  x  e.  A }  =  A
32eqeq1i 2047 . . 3  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
43exbii 1496 . 2  |-  ( E. x { x  |  x  e.  A }  =  { x }  <->  E. x  A  =  { x } )
51, 4bitri 173 1  |-  ( E! x  x  e.  A  <->  E. x  A  =  {
x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   E!weu 1900   {cab 2026   {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381
This theorem is referenced by: (None)
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