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Theorem uniss 4028
Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniss  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )

Proof of Theorem uniss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3334 . . . . 5  |-  ( A 
C_  B  ->  (
y  e.  A  -> 
y  e.  B ) )
21anim2d 549 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  y  /\  y  e.  A
)  ->  ( x  e.  y  /\  y  e.  B ) ) )
32eximdv 1632 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x  e.  y  /\  y  e.  A )  ->  E. y
( x  e.  y  /\  y  e.  B
) ) )
4 eluni 4010 . . 3  |-  ( x  e.  U. A  <->  E. y
( x  e.  y  /\  y  e.  A
) )
5 eluni 4010 . . 3  |-  ( x  e.  U. B  <->  E. y
( x  e.  y  /\  y  e.  B
) )
63, 4, 53imtr4g 262 . 2  |-  ( A 
C_  B  ->  (
x  e.  U. A  ->  x  e.  U. B
) )
76ssrdv 3346 1  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    e. wcel 1725    C_ wss 3312   U.cuni 4007
This theorem is referenced by:  unissi  4030  unissd  4031  intssuni2  4067  uniintsn  4079  relfld  5386  dffv2  5787  trcl  7653  cflm  8119  coflim  8130  cfslbn  8136  fin23lem41  8221  fin1a2lem12  8280  tskuni  8647  prdsval  13666  prdsbas  13668  prdsplusg  13669  prdsmulr  13670  prdsvsca  13671  prdshom  13677  mrcssv  13827  catcfuccl  14252  catcxpccl  14292  mrelatlub  14600  mreclat  14601  dprdres  15574  dmdprdsplit2lem  15591  tgcl  17022  distop  17048  fctop  17056  cctop  17058  neiptoptop  17183  cmpcld  17453  uncmp  17454  cmpfi  17459  bwth  17461  kgentopon  17558  txcmplem2  17662  filcon  17903  alexsubALTlem3  18068  alexsubALT  18070  ptcmplem3  18073  dyadmbllem  19479  shsupcl  22828  hsupss  22831  shatomistici  23852  cvmliftlem15  24973  frrlem5c  25542  comppfsc  26324  filnetlem3  26346  heiborlem1  26457  lssats  29649  lpssat  29650  lssatle  29652  lssat  29653  dicval  31813
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-v 2950  df-in 3319  df-ss 3326  df-uni 4008
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