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Theorem inex1 4057
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.
StepHypRef Expression
1 inex1.1 . . . 4  |-  A  e. 
_V
21zfauscl 4043 . . 3  |-  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) )
3 dfcleq 2131 . . . . 5  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  y  e.  ( A  i^i  B ) ) )
4 elin 3254 . . . . . . 7  |-  ( y  e.  ( A  i^i  B )  <->  ( y  e.  A  /\  y  e.  B ) )
54bibi2i 226 . . . . . 6  |-  ( ( y  e.  x  <->  y  e.  ( A  i^i  B ) )  <->  ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
65albii 1446 . . . . 5  |-  ( A. y ( y  e.  x  <->  y  e.  ( A  i^i  B ) )  <->  A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
73, 6bitri 183 . . . 4  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
87exbii 1584 . . 3  |-  ( E. x  x  =  ( A  i^i  B )  <->  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
92, 8mpbir 145 . 2  |-  E. x  x  =  ( A  i^i  B )
109issetri 2690 1  |-  ( A  i^i  B )  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   A.wal 1329    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2681    i^i cin 3065
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072
This theorem is referenced by:  inex2  4058  inex1g  4059  inuni  4075  bnd2  4092  peano5  4507  ssimaex  5475  ofmres  6027  tfrexlem  6224  elrest  12116  epttop  12248  tgrest  12327  resttopon  12329  restco  12332  cnrest2  12394  cnptopresti  12396  cnptoprest  12397  cnptoprest2  12398  txrest  12434
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