MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  n0moeu Structured version   Unicode version

Theorem n0moeu 3632
Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
Assertion
Ref Expression
n0moeu  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  E! x  x  e.  A
) )
Distinct variable group:    x, A

Proof of Theorem n0moeu
StepHypRef Expression
1 n0 3629 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
21biimpi 187 . . 3  |-  ( A  =/=  (/)  ->  E. x  x  e.  A )
32biantrurd 495 . 2  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
) )
4 eu5 2318 . 2  |-  ( E! x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
)
53, 4syl6bbr 255 1  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  E! x  x  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    e. wcel 1725   E!weu 2280   E*wmo 2281    =/= wne 2598   (/)c0 3620
This theorem is referenced by:  minveclem4a  19321  frg2wot1  28347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-nul 3621
  Copyright terms: Public domain W3C validator