Theorem inex1 4098

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 4084	. . 3 |- E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) )
3		dfcleq 2151	. . . . 5 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
4		elin 3290	. . . . . . 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 1450	. . . . 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 2721	1 |- ( A i^i B ) e. _V

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1333   = wceq 1335   E.wex 1472   e. wcel 2128   _Vcvv 2712   i^i cin 3101

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108

This theorem is referenced by:  inex2  4099  inex1g  4100  inuni  4116  bnd2  4134  peano5  4557  ssimaex  5529  ofmres  6084  tfrexlem  6281  elrest  12369  epttop  12501  tgrest  12580  resttopon  12582  restco  12585  cnrest2  12647  cnptopresti  12649  cnptoprest  12650  cnptoprest2  12651  txrest  12687