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Theorem unexb 4258
Description: Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
Assertion
Ref Expression
unexb  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A  u.  B )  e.  _V )

Proof of Theorem unexb
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, 6unex 4257 . . 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 ssexg 3970 . . . 4  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  A  e.  _V )
119, 10mpan 415 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  A  e.  _V )
12 ssun2 3162 . . . 4  |-  B  C_  ( A  u.  B
)
13 ssexg 3970 . . . 4  |-  ( ( B  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  B  e.  _V )
1412, 13mpan 415 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  B  e.  _V )
1511, 14jca 300 . 2  |-  ( ( A  u.  B )  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
168, 15impbii 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
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-sep 3949  ax-pr 4027  ax-un 4251
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
This theorem is referenced by:  unexg  4259  sucexb  4304  frecabex  6145
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