Theorem moeq 2994

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

Step	Hyp	Ref	Expression
1		isset 2822	. . . 4 |- ( A e. _V <-> E. x x = A )
2		eueq 2990	. . . 4 |- ( A e. _V <-> E! x x = A )
3	1, 2	bitr3i 186	. . 3 |- ( E. x x = A <-> E! x x = A )
4	3	biimpi 120	. 2 |- ( E. x x = A -> E! x x = A )
5		df-mo 2086	. 2 |- ( E* x x = A <-> ( E. x x = A -> E! x x = A ) )
6	4, 5	mpbir 146	1 |- E* x x = A

Syntax hints:   -> wi 4   = wceq 1398   E.wex 1541   E!weu 2082   E*wmo 2083   e. wcel 2205   _Vcvv 2815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817

This theorem is referenced by:  euxfr2dc  3004  reueq  3018  mosn  3727  sndisj  4107  disjxsn  4109  reusv1  4581  funopabeq  5390  funcnvsn  5403  fvmptg  5755  fvopab6  5776  ovmpt4g  6178  ovi3  6193  ov6g  6194  oprabex3  6324  1stconst  6419  2ndconst  6420  axaddf  8185  axmulf  8186