Theorem moeq 2854

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 2687	. . . 4 |- ( A e. _V <-> E. x x = A )
2		eueq 2850	. . . 4 |- ( A e. _V <-> E! x x = A )
3	1, 2	bitr3i 185	. . 3 |- ( E. x x = A <-> E! x x = A )
4	3	biimpi 119	. 2 |- ( E. x x = A -> E! x x = A )
5		df-mo 2001	. 2 |- ( E* x x = A <-> ( E. x x = A -> E! x x = A ) )
6	4, 5	mpbir 145	1 |- E* x x = A

Syntax hints:   -> wi 4   = wceq 1331   E.wex 1468   e. wcel 1480   E!weu 1997   E*wmo 1998   _Vcvv 2681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683

This theorem is referenced by:  euxfr2dc  2864  reueq  2878  mosn  3555  sndisj  3920  disjxsn  3922  reusv1  4374  funopabeq  5154  funcnvsn  5163  fvmptg  5490  fvopab6  5510  ovmpt4g  5886  ovi3  5900  ov6g  5901  oprabex3  6020  1stconst  6111  2ndconst  6112  axaddf  7669  axmulf  7670