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

Theorem unssi 3679
Description: An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
Hypotheses
Ref Expression
unssi.1  |-  A  C_  C
unssi.2  |-  B  C_  C
Assertion
Ref Expression
unssi  |-  ( A  u.  B )  C_  C

Proof of Theorem unssi
StepHypRef Expression
1 unssi.1 . . 3  |-  A  C_  C
2 unssi.2 . . 3  |-  B  C_  C
31, 2pm3.2i 455 . 2  |-  ( A 
C_  C  /\  B  C_  C )
4 unss 3678 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
53, 4mpbi 208 1  |-  ( A  u.  B )  C_  C
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    u. cun 3474    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-in 3483  df-ss 3490
This theorem is referenced by:  dmrnssfld  5261  tc2  8173  pwxpndom2  9043  ltrelxr  9648  nn0ssre  10799  nn0ssz  10885  dfle2  11353  difreicc  11652  hashxrcl  12397  ramxrcl  14394  strlemor1  14582  strleun  14585  cssincl  18514  leordtval2  19507  lecldbas  19514  aalioulem2  22491  taylfval  22516  axlowdimlem10  23958  konigsberg  24691  shunssji  25991  shsval3i  26010  shjshsi  26114  spanuni  26166  sshhococi  26168  esumcst  27739  hashf2  27758  sxbrsigalem3  27911  signswch  28186  mblfinlem3  29658  mblfinlem4  29659  comppfsc  29807  bj-unrab  33593  bj-tagss  33637  hdmapevec  36653
  Copyright terms: Public domain W3C validator