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Theorem unssd 3218
Description: A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
unssd.1  |-  ( ph  ->  A  C_  C )
unssd.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
unssd  |-  ( ph  ->  ( A  u.  B
)  C_  C )

Proof of Theorem unssd
StepHypRef Expression
1 unssd.1 . 2  |-  ( ph  ->  A  C_  C )
2 unssd.2 . 2  |-  ( ph  ->  B  C_  C )
3 unss 3216 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
43biimpi 119 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( A  u.  B
)  C_  C )
51, 2, 4syl2anc 406 1  |-  ( ph  ->  ( A  u.  B
)  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    u. cun 3035    C_ wss 3037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050
This theorem is referenced by:  tpssi  3652  casef  6925  un0addcl  8914  un0mulcl  8915  fzosplit  9847  fzouzsplit  9849  exmidunben  11784  strleund  11890  fsumcncntop  12542  bj-omtrans  12846
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