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Theorem bdunexb 11468
Description: Bounded version of unexb 4258. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdunex.bd1  |- BOUNDED  A
bdunex.bd2  |- BOUNDED  B
Assertion
Ref Expression
bdunexb  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A  u.  B )  e.  _V )

Proof of Theorem bdunexb
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3145 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21eleq1d 2156 . . 3  |-  ( x  =  A  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
3 uneq2 3146 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
43eleq1d 2156 . . 3  |-  ( y  =  B  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
5 vex 2622 . . . 4  |-  x  e. 
_V
6 vex 2622 . . . 4  |-  y  e. 
_V
75, 6bj-unex 11467 . . 3  |-  ( x  u.  y )  e. 
_V
82, 4, 7vtocl2g 2683 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  u.  B
)  e.  _V )
9 ssun1 3161 . . . 4  |-  A  C_  ( A  u.  B
)
10 bdunex.bd1 . . . . 5  |- BOUNDED  A
1110bdssexg 11452 . . . 4  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  A  e.  _V )
129, 11mpan 415 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  A  e.  _V )
13 ssun2 3162 . . . 4  |-  B  C_  ( A  u.  B
)
14 bdunex.bd2 . . . . 5  |- BOUNDED  B
1514bdssexg 11452 . . . 4  |-  ( ( B  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  B  e.  _V )
1613, 15mpan 415 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  B  e.  _V )
1712, 16jca 300 . 2  |-  ( ( A  u.  B )  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
188, 17impbii 124 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A  u.  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619    u. cun 2995    C_ wss 2997  BOUNDED wbdc 11388
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-pr 4027  ax-un 4251  ax-bd0 11361  ax-bdor 11364  ax-bdex 11367  ax-bdeq 11368  ax-bdel 11369  ax-bdsb 11370  ax-bdsep 11432
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-uni 3649  df-bdc 11389
This theorem is referenced by: (None)
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