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Theorem n0moeu 3742
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 3738 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
21biimpi 194 . . 3  |-  ( A  =/=  (/)  ->  E. x  x  e.  A )
32biantrurd 506 . 2  |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A  <->  ( E. x  x  e.  A  /\  E* x  x  e.  A )
) )
4 eu5 2260 . 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 367   E.wex 1627    e. wcel 1836   E!weu 2232   E*wmo 2233    =/= wne 2591   (/)c0 3728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-v 3053  df-dif 3409  df-nul 3729
This theorem is referenced by:  minveclem4a  21953  frg2wot1  25203
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