Theorem inex1 4223

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 4209	. . 3 |- E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) )
3		dfcleq 2225	. . . . 5 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
4		elin 3390	. . . . . . 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 1518	. . . . 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 1653	. . 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 2812	1 |- ( A i^i B ) e. _V

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1395   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   i^i cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  inex2  4224  inex1g  4225  inuni  4245  bnd2  4263  peano5  4696  ssimaex  5707  ofmres  6298  tfrexlem  6500  elrest  13347  epttop  14833  tgrest  14912  resttopon  14914  restco  14917  cnrest2  14979  cnptopresti  14981  cnptoprest  14982  cnptoprest2  14983  txrest  15019