MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eusn Structured version   Unicode version

Theorem eusn 3882
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 3878 . 2  |-  ( E! x  x  e.  A  <->  E. x { x  |  x  e.  A }  =  { x } )
2 abid2 2555 . . . 4  |-  { x  |  x  e.  A }  =  A
32eqeq1i 2445 . . 3  |-  ( { x  |  x  e.  A }  =  {
x }  <->  A  =  { x } )
43exbii 1593 . 2  |-  ( E. x { x  |  x  e.  A }  =  { x }  <->  E. x  A  =  { x } )
51, 4bitri 242 1  |-  ( E! x  x  e.  A  <->  E. x  A  =  {
x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2283   {cab 2424   {csn 3816
This theorem is referenced by:  reusv6OLD  4736  reusv7OLD  4737  funpartfv  25792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-sn 3822
  Copyright terms: Public domain W3C validator