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Theorem bdunexb 10854
Description: Bounded version of unexb 4197. (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 3120 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21eleq1d 2148 . . 3  |-  ( x  =  A  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
3 uneq2 3121 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
43eleq1d 2148 . . 3  |-  ( y  =  B  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
5 vex 2605 . . . 4  |-  x  e. 
_V
6 vex 2605 . . . 4  |-  y  e. 
_V
75, 6bj-unex 10853 . . 3  |-  ( x  u.  y )  e. 
_V
82, 4, 7vtocl2g 2663 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  u.  B
)  e.  _V )
9 ssun1 3136 . . . 4  |-  A  C_  ( A  u.  B
)
10 bdunex.bd1 . . . . 5  |- BOUNDED  A
1110bdssexg 10838 . . . 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 3137 . . . 4  |-  B  C_  ( A  u.  B
)
14 bdunex.bd2 . . . . 5  |- BOUNDED  B
1514bdssexg 10838 . . . 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 1285    e. wcel 1434   _Vcvv 2602    u. cun 2972    C_ wss 2974  BOUNDED wbdc 10774
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-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-pr 3966  ax-un 4190  ax-bd0 10747  ax-bdor 10750  ax-bdex 10753  ax-bdeq 10754  ax-bdel 10755  ax-bdsb 10756  ax-bdsep 10818
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-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-pr 3407  df-uni 3604  df-bdc 10775
This theorem is referenced by: (None)
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