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Theorem moeq 3102
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 2952 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 eueq 3098 . . . 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 2285 . 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 1550    = wceq 1652    e. wcel 1725   E!weu 2280   E*wmo 2281   _Vcvv 2948
This theorem is referenced by:  mosub  3104  euxfr2  3111  reueq  3123  sndisj  4196  disjxsn  4198  reusv1  4715  reusv2lem1  4716  reuxfr2d  4738  funopabeq  5479  funcnvsn  5488  fvmptg  5796  fvopab6  5818  oprabex3  6180  ovmpt4g  6188  ov3  6202  ov6g  6203  1stconst  6427  2ndconst  6428  iunmapdisj  7896  axaddf  9012  axmulf  9013  reuxfr3d  23968  abrexdom2jm  23981  abrexdom2  26424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
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