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Theorem unexb 4205
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 3118 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21eleq1d 2122 . . 3  |-  ( x  =  A  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
3 uneq2 3119 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
43eleq1d 2122 . . 3  |-  ( y  =  B  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
5 vex 2577 . . . 4  |-  x  e. 
_V
6 vex 2577 . . . 4  |-  y  e. 
_V
75, 6unex 4204 . . 3  |-  ( x  u.  y )  e. 
_V
82, 4, 7vtocl2g 2634 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  u.  B
)  e.  _V )
9 ssun1 3134 . . . 4  |-  A  C_  ( A  u.  B
)
10 ssexg 3924 . . . 4  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  A  e.  _V )
119, 10mpan 408 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  A  e.  _V )
12 ssun2 3135 . . . 4  |-  B  C_  ( A  u.  B
)
13 ssexg 3924 . . . 4  |-  ( ( B  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  B  e.  _V )
1412, 13mpan 408 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  B  e.  _V )
1511, 14jca 294 . 2  |-  ( ( A  u.  B )  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
168, 15impbii 121 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A  u.  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574    u. cun 2943    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-uni 3609
This theorem is referenced by:  unexg  4206  sucexb  4251  frecabex  6015
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