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Theorem bdinex1 13202
Description: Bounded version of inex1 4062. (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 13149 . . . . 5  |- BOUNDED  y  e.  B
43bdzfauscl 13193 . . . 4  |-  ( A  e.  _V  ->  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
51, 4ax-mp 5 . . 3  |-  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) )
6 dfcleq 2133 . . . . 5  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  y  e.  ( A  i^i  B ) ) )
7 elin 3259 . . . . . . 7  |-  ( y  e.  ( A  i^i  B )  <->  ( y  e.  A  /\  y  e.  B ) )
87bibi2i 226 . . . . . 6  |-  ( ( y  e.  x  <->  y  e.  ( A  i^i  B ) )  <->  ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
98albii 1446 . . . . 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 183 . . . 4  |-  ( x  =  ( A  i^i  B )  <->  A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
1110exbii 1584 . . 3  |-  ( E. x  x  =  ( A  i^i  B )  <->  E. x A. y ( y  e.  x  <->  ( y  e.  A  /\  y  e.  B ) ) )
125, 11mpbir 145 . 2  |-  E. x  x  =  ( A  i^i  B )
1312issetri 2695 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 2686    i^i cin 3070  BOUNDED wbdc 13143
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 2121  ax-bdsep 13187
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-bdc 13144
This theorem is referenced by:  bdinex2  13203  bdinex1g  13204  bdpeano5  13246
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