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Theorem bdunexb 14321
Description: Bounded version of unexb 4439. (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 3282 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21eleq1d 2246 . . 3  |-  ( x  =  A  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
3 uneq2 3283 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
43eleq1d 2246 . . 3  |-  ( y  =  B  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
5 vex 2740 . . . 4  |-  x  e. 
_V
6 vex 2740 . . . 4  |-  y  e. 
_V
75, 6bj-unex 14320 . . 3  |-  ( x  u.  y )  e. 
_V
82, 4, 7vtocl2g 2801 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  u.  B
)  e.  _V )
9 ssun1 3298 . . . 4  |-  A  C_  ( A  u.  B
)
10 bdunex.bd1 . . . . 5  |- BOUNDED  A
1110bdssexg 14305 . . . 4  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  A  e.  _V )
129, 11mpan 424 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  A  e.  _V )
13 ssun2 3299 . . . 4  |-  B  C_  ( A  u.  B
)
14 bdunex.bd2 . . . . 5  |- BOUNDED  B
1514bdssexg 14305 . . . 4  |-  ( ( B  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  B  e.  _V )
1613, 15mpan 424 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  B  e.  _V )
1712, 16jca 306 . 2  |-  ( ( A  u.  B )  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
188, 17impbii 126 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A  u.  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737    u. cun 3127    C_ wss 3129  BOUNDED wbdc 14241
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-13 2150  ax-14 2151  ax-ext 2159  ax-pr 4206  ax-un 4430  ax-bd0 14214  ax-bdor 14217  ax-bdex 14220  ax-bdeq 14221  ax-bdel 14222  ax-bdsb 14223  ax-bdsep 14285
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-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-uni 3808  df-bdc 14242
This theorem is referenced by: (None)
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