MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unss12 Structured version   Unicode version

Theorem unss12 3511
Description: Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
Assertion
Ref Expression
unss12  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  u.  C
)  C_  ( B  u.  D ) )

Proof of Theorem unss12
StepHypRef Expression
1 unss1 3508 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 unss2 3510 . 2  |-  ( C 
C_  D  ->  ( B  u.  C )  C_  ( B  u.  D
) )
31, 2sylan9ss 3353 1  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  u.  C
)  C_  ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    u. cun 3310    C_ wss 3312
This theorem is referenced by:  pwssun  4481  fun  5599  undom  7188  finsschain  7405  dprd2da  15592  dmdprdsplit2lem  15595  lspun  16055  spanuni  23038  sshhococi  23040  mblfinlem2  26235  mvdco  27356  dochdmj1  32125
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-un 3317  df-in 3319  df-ss 3326
  Copyright terms: Public domain W3C validator