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Theorem inex1 4132
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 4118 . . 3  |-  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) )
3 dfcleq 2169 . . . . 5  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  y  e.  ( A  i^i  B ) ) )
4 elin 3316 . . . . . . 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 1468 . . . . 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 1603 . . 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 2744 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 1490    e. wcel 2146   _Vcvv 2735    i^i cin 3126
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-sep 4116
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133
This theorem is referenced by:  inex2  4133  inex1g  4134  inuni  4150  bnd2  4168  peano5  4591  ssimaex  5569  ofmres  6127  tfrexlem  6325  elrest  12616  epttop  13083  tgrest  13162  resttopon  13164  restco  13167  cnrest2  13229  cnptopresti  13231  cnptoprest  13232  cnptoprest2  13233  txrest  13269
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