Theorem moeq 2948

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 2778	. . . 4 |- ( A e. _V <-> E. x x = A )
2		eueq 2944	. . . 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 2058	. 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 1373   E.wex 1515   E!weu 2054   E*wmo 2055   e. wcel 2176   _Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  euxfr2dc  2958  reueq  2972  mosn  3669  sndisj  4041  disjxsn  4043  reusv1  4506  funopabeq  5308  funcnvsn  5320  fvmptg  5657  fvopab6  5678  ovmpt4g  6070  ovi3  6085  ov6g  6086  oprabex3  6216  1stconst  6309  2ndconst  6310  axaddf  7983  axmulf  7984