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

Theorem unss 3485
Description: The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
Assertion
Ref Expression
unss  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)

Proof of Theorem unss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3301 . 2  |-  ( ( A  u.  B ) 
C_  C  <->  A. x
( x  e.  ( A  u.  B )  ->  x  e.  C
) )
2 19.26 1600 . . 3  |-  ( A. x ( ( x  e.  A  ->  x  e.  C )  /\  (
x  e.  B  ->  x  e.  C )
)  <->  ( A. x
( x  e.  A  ->  x  e.  C )  /\  A. x ( x  e.  B  ->  x  e.  C )
) )
3 elun 3452 . . . . . 6  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
43imbi1i 316 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  ->  x  e.  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  ->  x  e.  C ) )
5 jaob 759 . . . . 5  |-  ( ( ( x  e.  A  \/  x  e.  B
)  ->  x  e.  C )  <->  ( (
x  e.  A  ->  x  e.  C )  /\  ( x  e.  B  ->  x  e.  C ) ) )
64, 5bitri 241 . . . 4  |-  ( ( x  e.  ( A  u.  B )  ->  x  e.  C )  <->  ( ( x  e.  A  ->  x  e.  C )  /\  ( x  e.  B  ->  x  e.  C ) ) )
76albii 1572 . . 3  |-  ( A. x ( x  e.  ( A  u.  B
)  ->  x  e.  C )  <->  A. x
( ( x  e.  A  ->  x  e.  C )  /\  (
x  e.  B  ->  x  e.  C )
) )
8 dfss2 3301 . . . 4  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
9 dfss2 3301 . . . 4  |-  ( B 
C_  C  <->  A. x
( x  e.  B  ->  x  e.  C ) )
108, 9anbi12i 679 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A. x ( x  e.  A  ->  x  e.  C )  /\  A. x ( x  e.  B  ->  x  e.  C ) ) )
112, 7, 103bitr4i 269 . 2  |-  ( A. x ( x  e.  ( A  u.  B
)  ->  x  e.  C )  <->  ( A  C_  C  /\  B  C_  C ) )
121, 11bitr2i 242 1  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   A.wal 1546    e. wcel 1721    u. cun 3282    C_ wss 3284
This theorem is referenced by:  unssi  3486  unssd  3487  unssad  3488  unssbd  3489  nsspssun  3538  uneqin  3556  uneqdifeq  3680  prss  3916  prssg  3917  ssunsn2  3922  tpss  3928  pwundif  4454  eqrelrel  4940  xpsspw  4949  xpsspwOLD  4950  relun  4954  relcoi2  5360  fnsuppres  5915  dfer2  6869  isinf  7285  fiin  7389  trcl  7624  supxrun  10854  isumltss  12587  rpnnen2  12784  lubun  14509  isipodrs  14546  fpwipodrs  14549  ipodrsima  14550  aspval2  16364  unocv  16866  uncld  17064  restntr  17204  cmpcld  17423  uncmp  17424  ufprim  17898  tsmsfbas  18114  ovolctb2  19345  ovolun  19352  unmbl  19389  plyun0  20073  sshjcl  22814  sshjval2  22870  shlub  22873  ssjo  22906  spanuni  23003  dfon2lem3  25359  dfon2lem7  25363  wfrlem15  25488  mblfinlem2  26148  ismblfin  26150  clsun  26225  lsmfgcl  27044  lubunNEW  29460  paddssat  30300  pclunN  30384  paddunN  30413  poldmj1N  30414  pclfinclN  30436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-un 3289  df-in 3291  df-ss 3298
  Copyright terms: Public domain W3C validator