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Theorem inex1 4139
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 4125 . . 3  |-  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) )
3 dfcleq 2171 . . . . 5  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  y  e.  ( A  i^i  B ) ) )
4 elin 3320 . . . . . . 7  |-  ( y  e.  ( A  i^i  B )  <->  ( y  e.  A  /\  y  e.  B ) )
54bibi2i 227 . . . . . 6  |-  ( ( y  e.  x  <->  y  e.  ( A  i^i  B ) )  <->  ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
65albii 1470 . . . . 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 184 . . . 4  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
87exbii 1605 . . 3  |-  ( E. x  x  =  ( A  i^i  B )  <->  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
92, 8mpbir 146 . 2  |-  E. x  x  =  ( A  i^i  B )
109issetri 2748 1  |-  ( A  i^i  B )  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   _Vcvv 2739    i^i cin 3130
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137
This theorem is referenced by:  inex2  4140  inex1g  4141  inuni  4157  bnd2  4175  peano5  4599  ssimaex  5579  ofmres  6139  tfrexlem  6337  elrest  12700  epttop  13629  tgrest  13708  resttopon  13710  restco  13713  cnrest2  13775  cnptopresti  13777  cnptoprest  13778  cnptoprest2  13779  txrest  13815
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