MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unss2 Unicode version

Theorem unss2 3434
Description: Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
unss2  |-  ( A 
C_  B  ->  ( C  u.  A )  C_  ( C  u.  B
) )

Proof of Theorem unss2
StepHypRef Expression
1 unss1 3432 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 uncom 3407 . 2  |-  ( C  u.  A )  =  ( A  u.  C
)
3 uncom 3407 . 2  |-  ( C  u.  B )  =  ( B  u.  C
)
41, 2, 33sstr4g 3305 1  |-  ( A 
C_  B  ->  ( C  u.  A )  C_  ( C  u.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    u. cun 3236    C_ wss 3238
This theorem is referenced by:  unss12  3435  ord3ex  4302  xpider  6872  fin1a2lem13  8185  canthp1lem2  8422  uniioombllem3  19155  volcn  19176  dvres2lem  19475  bnj1413  28817  bnj1408  28818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-un 3243  df-in 3245  df-ss 3252
  Copyright terms: Public domain W3C validator