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Theorem bdinex1 10848
Description: Bounded version of inex1 3920. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdinex1.bd  |- BOUNDED  B
bdinex1.1  |-  A  e. 
_V
Assertion
Ref Expression
bdinex1  |-  ( A  i^i  B )  e. 
_V

Proof of Theorem bdinex1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdinex1.1 . . . 4  |-  A  e. 
_V
2 bdinex1.bd . . . . . 6  |- BOUNDED  B
32bdeli 10795 . . . . 5  |- BOUNDED  y  e.  B
43bdzfauscl 10839 . . . 4  |-  ( A  e.  _V  ->  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
51, 4ax-mp 7 . . 3  |-  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) )
6 dfcleq 2076 . . . . 5  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  y  e.  ( A  i^i  B ) ) )
7 elin 3156 . . . . . . 7  |-  ( y  e.  ( A  i^i  B )  <->  ( y  e.  A  /\  y  e.  B ) )
87bibi2i 225 . . . . . 6  |-  ( ( y  e.  x  <->  y  e.  ( A  i^i  B ) )  <->  ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
98albii 1400 . . . . 5  |-  ( A. y ( y  e.  x  <->  y  e.  ( A  i^i  B ) )  <->  A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
106, 9bitri 182 . . . 4  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
1110exbii 1537 . . 3  |-  ( E. x  x  =  ( A  i^i  B )  <->  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
125, 11mpbir 144 . 2  |-  E. x  x  =  ( A  i^i  B )
1312issetri 2609 1  |-  ( A  i^i  B )  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602    i^i cin 2973  BOUNDED wbdc 10789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-bdsep 10833
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-bdc 10790
This theorem is referenced by:  bdinex2  10849  bdinex1g  10850  bdpeano5  10896
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