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Theorem unss 3250
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 3086 . 2  |-  ( ( A  u.  B ) 
C_  C  <->  A. x
( x  e.  ( A  u.  B )  ->  x  e.  C
) )
2 19.26 1457 . . 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 3217 . . . . . 6  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
43imbi1i 237 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  ->  x  e.  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  ->  x  e.  C ) )
5 jaob 699 . . . . 5  |-  ( ( ( x  e.  A  \/  x  e.  B
)  ->  x  e.  C )  <->  ( (
x  e.  A  ->  x  e.  C )  /\  ( x  e.  B  ->  x  e.  C ) ) )
64, 5bitri 183 . . . 4  |-  ( ( x  e.  ( A  u.  B )  ->  x  e.  C )  <->  ( ( x  e.  A  ->  x  e.  C )  /\  ( x  e.  B  ->  x  e.  C ) ) )
76albii 1446 . . 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 3086 . . . 4  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
9 dfss2 3086 . . . 4  |-  ( B 
C_  C  <->  A. x
( x  e.  B  ->  x  e.  C ) )
108, 9anbi12i 455 . . 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 211 . 2  |-  ( A. x ( x  e.  ( A  u.  B
)  ->  x  e.  C )  <->  ( A  C_  C  /\  B  C_  C ) )
121, 11bitr2i 184 1  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697   A.wal 1329    e. wcel 1480    u. cun 3069    C_ wss 3071
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084
This theorem is referenced by:  unssi  3251  unssd  3252  unssad  3253  unssbd  3254  uneqin  3327  undifss  3443  prss  3676  prssg  3677  tpss  3685  pwundifss  4207  ordsucss  4420  elnn  4519  eqrelrel  4640  xpsspw  4651  relun  4656  relcoi2  5069  dfer2  6430  fimaxre2  11010  uncld  12296  bdeqsuc  13163  exmid1stab  13279
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