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Theorem unexb 4415
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 3265 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21eleq1d 2233 . . 3  |-  ( x  =  A  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
3 uneq2 3266 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
43eleq1d 2233 . . 3  |-  ( y  =  B  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
5 vex 2725 . . . 4  |-  x  e. 
_V
6 vex 2725 . . . 4  |-  y  e. 
_V
75, 6unex 4414 . . 3  |-  ( x  u.  y )  e. 
_V
82, 4, 7vtocl2g 2786 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  u.  B
)  e.  _V )
9 ssun1 3281 . . . 4  |-  A  C_  ( A  u.  B
)
10 ssexg 4116 . . . 4  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  A  e.  _V )
119, 10mpan 421 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  A  e.  _V )
12 ssun2 3282 . . . 4  |-  B  C_  ( A  u.  B
)
13 ssexg 4116 . . . 4  |-  ( ( B  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  B  e.  _V )
1412, 13mpan 421 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  B  e.  _V )
1511, 14jca 304 . 2  |-  ( ( A  u.  B )  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
168, 15impbii 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 1342    e. wcel 2135   _Vcvv 2722    u. cun 3110    C_ wss 3112
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pr 4182  ax-un 4406
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-sn 3577  df-pr 3578  df-uni 3785
This theorem is referenced by:  unexg  4416  sucexb  4469  frecabex  6358  djuexb  7001
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