Theorem moeq 1431

Description: There is at most one set equal to a class.

Assertion

Ref	Expression
moeq	|- E*x x = A

Distinct variable group(s):   x,A

Proof of Theorem moeq

Step	Hyp	Ref	Expression
1		isset 1351	. . . 4 |- (A e. V <-> E.x x = A)
2		eueq 1427	. . . 4 |- (A e. V <-> E!x x = A)
3	1, 2	bitr3 153	. . 3 |- (E.x x = A <-> E!x x = A)
4	3	biimp 133	. 2 |- (E.x x = A -> E!x x = A)
5		df-mo 1010	. 2 |- (E*x x = A <-> (E.x x = A -> E!x x = A))
6	4, 5	mpbir 165	1 |- E*x x = A

Syntax hints:   -> wi 2  E.wex 678  E!weu 1007  E*wmo 1008   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  mosub 1433  euxfr2 1435  reuxfr2 1579  funopabeq 2695  fconst 2774  fvex 2838  fvopab4g 2870  oprabex2 3045  oprabex3 3046  oprabval2g 3050  oprabval3 3052  qsex 3231  pw2en 3348  mapxpen 3390  xpmapenlem2 3392  xpmapenlem4 3394  xpmapenlem5 3395  aceq4 3557  aceq6a 3564  seqval 4665  occllem6 5185  occllem7 5186  projlem25 5217  projlem26 5218  projlem31 5223  pjmvalt 5245

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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