Theorem inex1 4070

Description: Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
inex1.1	|- A e. _V

Assertion

Ref	Expression
inex1	|- ( A i^i B ) e. _V

Proof of Theorem inex1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inex1.1	. . . 4 |- A e. _V
2	1	zfauscl 4056	. . 3 |- E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) )
3		dfcleq 2134	. . . . 5 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
4		elin 3264	. . . . . . 7 |- ( y e. ( A i^i B ) <-> ( y e. A /\ y e. B ) )
5	4	bibi2i 226	. . . . . 6 |- ( ( y e. x <-> y e. ( A i^i B ) ) <-> ( y e. x <-> ( y e. A /\ y e. B ) ) )
6	5	albii 1447	. . . . 5 |- ( A. y ( y e. x <-> y e. ( A i^i B ) ) <-> A. y ( y e. x <-> ( y e. A /\ y e. B ) ) )
7	3, 6	bitri 183	. . . 4 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> ( y e. A /\ y e. B ) ) )
8	7	exbii 1585	. . 3 |- ( E. x x = ( A i^i B ) <-> E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) ) )
9	2, 8	mpbir 145	. 2 |- E. x x = ( A i^i B )
10	9	issetri 2698	1 |- ( A i^i B ) e. _V

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1330   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   i^i cin 3075

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082

This theorem is referenced by:  inex2  4071  inex1g  4072  inuni  4088  bnd2  4105  peano5  4520  ssimaex  5490  ofmres  6042  tfrexlem  6239  elrest  12166  epttop  12298  tgrest  12377  resttopon  12379  restco  12382  cnrest2  12444  cnptopresti  12446  cnptoprest  12447  cnptoprest2  12448  txrest  12484