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Theorem unssi 3282
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 270 . 2  |-  ( A 
C_  C  /\  B  C_  C )
4 unss 3281 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
53, 4mpbi 144 1  |-  ( A  u.  B )  C_  C
Colors of variables: wff set class
Syntax hints:    /\ wa 103    u. cun 3100    C_ wss 3102
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115
This theorem is referenced by:  undifabs  3470  inundifss  3471  dmrnssfld  4849  caserel  7031  ltrelxr  7938  nn0ssre  9094  nn0ssz  9185  strleun  12290
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