Theorem moeq 1911

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

Assertion

Ref	Expression
moeq	|- E*x x = A

Distinct variable group:   x,A

Proof of Theorem moeq

Step	Hyp	Ref	Expression
1		isset 1805	. . . 4 |- (A e. V <-> E.x x = A)
2		eueq 1907	. . . 4 |- (A e. V <-> E!x x = A)
3	1, 2	bitr3 175	. . 3 |- (E.x x = A <-> E!x x = A)
4	3	biimp 151	. 2 |- (E.x x = A -> E!x x = A)
5		df-mo 1376	. 2 |- (E*x x = A <-> (E.x x = A -> E!x x = A))
6	4, 5	mpbir 190	1 |- E*x x = A

Syntax hints:   -> wi 3   = wceq 953   e. wcel 955  E.wex 977  E!weu 1373  E*wmo 1374  Vcvv 1802

This theorem is referenced by:  mosub 1913  euxfr2 1916  reuxfr2 2893  funopabeq 3535  opabex2 3596  opabex2g 3597  fconst 3643  fvex 3717  fvopab4g 3764  oprabex2g 4005  oprabex3 4007  oprabval2gf 4011  oprabval3 4015  oprabval6g 4017  2ndconst 4081  axaddopr 5237  axmulopr 5238  spwval2 8577

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

Copyright terms: Public domain