MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unexb Structured version   Unicode version

Theorem unexb 4701
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 3486 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21eleq1d 2501 . . 3  |-  ( x  =  A  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
3 uneq2 3487 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
43eleq1d 2501 . . 3  |-  ( y  =  B  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
5 vex 2951 . . . 4  |-  x  e. 
_V
6 vex 2951 . . . 4  |-  y  e. 
_V
75, 6unex 4699 . . 3  |-  ( x  u.  y )  e. 
_V
82, 4, 7vtocl2g 3007 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  u.  B
)  e.  _V )
9 ssun1 3502 . . . 4  |-  A  C_  ( A  u.  B
)
10 ssexg 4341 . . . 4  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  A  e.  _V )
119, 10mpan 652 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  A  e.  _V )
12 ssun2 3503 . . . 4  |-  B  C_  ( A  u.  B
)
13 ssexg 4341 . . . 4  |-  ( ( B  C_  ( A  u.  B )  /\  ( A  u.  B )  e.  _V )  ->  B  e.  _V )
1412, 13mpan 652 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  B  e.  _V )
1511, 14jca 519 . 2  |-  ( ( A  u.  B )  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
168, 15impbii 181 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A  u.  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310    C_ wss 3312
This theorem is referenced by:  unexg  4702  sucexb  4781  fodomr  7250  cdaval  8042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813  df-uni 4008
  Copyright terms: Public domain W3C validator