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Theorem n0moeu 3736
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 3732 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
21biimpi 199 . . 3  |-  ( A  =/=  (/)  ->  E. x  x  e.  A )
32biantrurd 516 . 2  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
) )
4 eu5 2345 . 2  |-  ( E! x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
)
53, 4syl6bbr 271 1  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  E! x  x  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   E.wex 1671    e. wcel 1904   E!weu 2319   E*wmo 2320    =/= wne 2641   (/)c0 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-nul 3723
This theorem is referenced by:  minveclem4a  22450  minveclem4aOLD  22462  frg2wot1  25864
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