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Theorem n0moeu 3759
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 3755 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
21biimpi 194 . . 3  |-  ( A  =/=  (/)  ->  E. x  x  e.  A )
32biantrurd 508 . 2  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
) )
4 eu5 2292 . 2  |-  ( E! x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
)
53, 4syl6bbr 263 1  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  E! x  x  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1587    e. wcel 1758   E!weu 2262   E*wmo 2263    =/= wne 2648   (/)c0 3746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-dif 3440  df-nul 3747
This theorem is referenced by:  minveclem4a  21050  frg2wot1  30799
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