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Theorem moeq 3046
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 2896 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 eueq 3042 . . . 4  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2bitr3i 243 . . 3  |-  ( E. x  x  =  A  <-> 
E! x  x  =  A )
43biimpi 187 . 2  |-  ( E. x  x  =  A  ->  E! x  x  =  A )
5 df-mo 2236 . 2  |-  ( E* x  x  =  A  <-> 
( E. x  x  =  A  ->  E! x  x  =  A
) )
64, 5mpbir 201 1  |-  E* x  x  =  A
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1717   E!weu 2231   E*wmo 2232   _Vcvv 2892
This theorem is referenced by:  mosub  3048  euxfr2  3055  reueq  3067  sndisj  4138  disjxsn  4140  reusv1  4656  reusv2lem1  4657  reuxfr2d  4679  funopabeq  5420  funcnvsn  5429  fvmptg  5736  fvopab6  5758  oprabex3  6120  ovmpt4g  6128  ov3  6142  ov6g  6143  1stconst  6367  2ndconst  6368  iunmapdisj  7830  axaddf  8946  axmulf  8947  reuxfr3d  23813  abrexdom2jm  23826  abrexdom2  26117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-v 2894
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