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Theorem moeq 3279
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 3117 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 eueq 3275 . . . 4  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2bitr3i 251 . . 3  |-  ( E. x  x  =  A  <-> 
E! x  x  =  A )
43biimpi 194 . 2  |-  ( E. x  x  =  A  ->  E! x  x  =  A )
5 df-mo 2280 . 2  |-  ( E* x  x  =  A  <-> 
( E. x  x  =  A  ->  E! x  x  =  A
) )
64, 5mpbir 209 1  |-  E* x  x  =  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   E*wmo 2276   _Vcvv 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115
This theorem is referenced by:  mosub  3281  euxfr2  3288  reueq  3301  sndisj  4439  disjxsn  4441  reusv1  4647  reusv2lem1  4648  reuxfr2d  4670  funopabeq  5620  funcnvsn  5631  fvmptg  5946  fvopab6  5972  ovmpt4g  6407  ov3  6421  ov6g  6422  oprabex3  6770  1stconst  6868  2ndconst  6869  iunmapdisj  8400  axaddf  9518  axmulf  9519  joinfval  15481  joinval  15485  meetfval  15495  meetval  15499  reuxfr3d  27061  abrexdom2jm  27077  abrexdom2  29823
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