Theorem moeq 2716

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 2561	. . . 4 |- ( A e. _V <-> E. x x = A )
2		eueq 2712	. . . 4 |- ( A e. _V <-> E! x x = A )
3	1, 2	bitr3i 175	. . 3 |- ( E. x x = A <-> E! x x = A )
4	3	biimpi 113	. 2 |- ( E. x x = A -> E! x x = A )
5		df-mo 1904	. 2 |- ( E* x x = A <-> ( E. x x = A -> E! x x = A ) )
6	4, 5	mpbir 134	1 |- E* x x = A

Syntax hints:   -> wi 4   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   E*wmo 1901   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  euxfr2dc  2726  reueq  2738  mosn  3406  sndisj  3760  disjxsn  3762  reusv1  4190  funopabeq  4936  funcnvsn  4945  fvmptg  5248  fvopab6  5264  ovmpt4g  5623  ovi3  5637  ov6g  5638  oprabex3  5756  1stconst  5842  2ndconst  5843