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Theorem moeq 2954
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 2805 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 eueq 2950 . . . 4  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2bitr3i 242 . . 3  |-  ( E. x  x  =  A  <-> 
E! x  x  =  A )
43biimpi 186 . 2  |-  ( E. x  x  =  A  ->  E! x  x  =  A )
5 df-mo 2161 . 2  |-  ( E* x  x  =  A  <-> 
( E. x  x  =  A  ->  E! x  x  =  A
) )
64, 5mpbir 200 1  |-  E* x  x  =  A
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   E*wmo 2157   _Vcvv 2801
This theorem is referenced by:  mosub  2956  euxfr2  2963  reueq  2975  sndisj  4031  disjxsn  4033  reusv1  4550  reusv2lem1  4551  reuxfr2d  4573  funopabeq  5304  funcnvsn  5313  fvmptg  5616  fvopab6  5637  oprabex3  5978  ovmpt4g  5986  ov3  6000  ov6g  6001  1stconst  6223  2ndconst  6224  iunmapdisj  7666  axaddf  8783  axmulf  8784  reuxfr3d  23154  abrexdom2jm  23182  cmp2morp  26061  cmpmorfun  26074  abrexdom2  26509
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-v 2803
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