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Theorem n0moeu 3480
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 3477 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
21biimpi 186 . . 3  |-  ( A  =/=  (/)  ->  E. x  x  e.  A )
32biantrurd 494 . 2  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
) )
4 eu5 2194 . 2  |-  ( E! x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
)
53, 4syl6bbr 254 1  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  E! x  x  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    e. wcel 1696   E!weu 2156   E*wmo 2157    =/= wne 2459   (/)c0 3468
This theorem is referenced by:  minveclem4a  18810  nolimf2  25723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-nul 3469
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