Theorem inex1 4152

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 4138	. . 3 |- E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) )
3		dfcleq 2183	. . . . 5 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
4		elin 3333	. . . . . . 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 1481	. . . . 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 1616	. . 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 2761	1 |- ( A i^i B ) e. _V

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2160   _Vcvv 2752   i^i cin 3143

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150

This theorem is referenced by:  inex2  4153  inex1g  4154  inuni  4170  bnd2  4188  peano5  4612  ssimaex  5593  ofmres  6155  tfrexlem  6353  elrest  12717  epttop  13974  tgrest  14053  resttopon  14055  restco  14058  cnrest2  14120  cnptopresti  14122  cnptoprest  14123  cnptoprest2  14124  txrest  14160