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Theorem unssi 3524
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 443 . 2  |-  ( A 
C_  C  /\  B  C_  C )
4 unss 3523 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
53, 4mpbi 201 1  |-  ( A  u.  B )  C_  C
Colors of variables: wff set class
Syntax hints:    /\ wa 360    u. cun 3320    C_ wss 3322
This theorem is referenced by:  dmrnssfld  5132  tc2  7684  pwxpndom2  8545  ltrelxr  9144  nn0ssre  10230  nn0ssz  10307  dfle2  10745  difreicc  11033  hashxrcl  11645  ramxrcl  13390  strlemor1  13561  strleun  13564  cssincl  16920  leordtval2  17281  lecldbas  17288  aalioulem2  20255  taylfval  20280  konigsberg  21714  shunssji  22876  shsval3i  22895  shjshsi  22999  spanuni  23051  sshhococi  23053  esumcst  24460  hashf2  24479  sxbrsigalem3  24627  axlowdimlem10  25895  mblfinlem3  26257  mblfinlem4  26258  comppfsc  26401  hdmapevec  32710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-in 3329  df-ss 3336
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