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Theorem bdunexb 13045
Description: Bounded version of unexb 4333. (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 3193 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21eleq1d 2186 . . 3  |-  ( x  =  A  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
3 uneq2 3194 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
43eleq1d 2186 . . 3  |-  ( y  =  B  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
5 vex 2663 . . . 4  |-  x  e. 
_V
6 vex 2663 . . . 4  |-  y  e. 
_V
75, 6bj-unex 13044 . . 3  |-  ( x  u.  y )  e. 
_V
82, 4, 7vtocl2g 2724 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  u.  B
)  e.  _V )
9 ssun1 3209 . . . 4  |-  A  C_  ( A  u.  B
)
10 bdunex.bd1 . . . . 5  |- BOUNDED  A
1110bdssexg 13029 . . . 4  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  A  e.  _V )
129, 11mpan 420 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  A  e.  _V )
13 ssun2 3210 . . . 4  |-  B  C_  ( A  u.  B
)
14 bdunex.bd2 . . . . 5  |- BOUNDED  B
1514bdssexg 13029 . . . 4  |-  ( ( B  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  B  e.  _V )
1613, 15mpan 420 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  B  e.  _V )
1712, 16jca 304 . 2  |-  ( ( A  u.  B )  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
188, 17impbii 125 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A  u.  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660    u. cun 3039    C_ wss 3041  BOUNDED wbdc 12965
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdor 12941  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-uni 3707  df-bdc 12966
This theorem is referenced by: (None)
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