Theorem inex1 4218

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 4204	. . 3 |- E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) )
3		dfcleq 2223	. . . . 5 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
4		elin 3387	. . . . . . 7 |- ( y e. ( A i^i B ) <-> ( y e. A /\ y e. B ) )
5	4	bibi2i 227	. . . . . 6 |- ( ( y e. x <-> y e. ( A i^i B ) ) <-> ( y e. x <-> ( y e. A /\ y e. B ) ) )
6	5	albii 1516	. . . . 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 184	. . . 4 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> ( y e. A /\ y e. B ) ) )
8	7	exbii 1651	. . 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 146	. 2 |- E. x x = ( A i^i B )
10	9	issetri 2809	1 |- ( A i^i B ) e. _V

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   i^i cin 3196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4202

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203

This theorem is referenced by:  inex2  4219  inex1g  4220  inuni  4239  bnd2  4257  peano5  4690  ssimaex  5695  ofmres  6281  tfrexlem  6480  elrest  13279  epttop  14764  tgrest  14843  resttopon  14845  restco  14848  cnrest2  14910  cnptopresti  14912  cnptoprest  14913  cnptoprest2  14914  txrest  14950