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Theorem moeq 2768
Description: There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
Assertion
Ref Expression
moeq  |-  E* x  x  =  A
Distinct variable group:    x, A

Proof of Theorem moeq
StepHypRef Expression
1 isset 2606 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 eueq 2764 . . . 4  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2bitr3i 184 . . 3  |-  ( E. x  x  =  A  <-> 
E! x  x  =  A )
43biimpi 118 . 2  |-  ( E. x  x  =  A  ->  E! x  x  =  A )
5 df-mo 1946 . 2  |-  ( E* x  x  =  A  <-> 
( E. x  x  =  A  ->  E! x  x  =  A
) )
64, 5mpbir 144 1  |-  E* x  x  =  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1942   E*wmo 1943   _Vcvv 2602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by:  euxfr2dc  2778  reueq  2790  mosn  3437  sndisj  3789  disjxsn  3791  reusv1  4216  funopabeq  4966  funcnvsn  4975  fvmptg  5280  fvopab6  5296  ovmpt4g  5654  ovi3  5668  ov6g  5669  oprabex3  5787  1stconst  5873  2ndconst  5874
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