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Theorem trunitr 25109
Description: The union of two transitive classes is transitive. JFM CLASSES1. th. 55 (Contributed by FL, 16-Apr-2011.)
Assertion
Ref Expression
trunitr  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  u.  B
) )

Proof of Theorem trunitr
StepHypRef Expression
1 uniun 3846 . . 3  |-  U. ( A  u.  B )  =  ( U. A  u.  U. B )
2 df-tr 4114 . . . 4  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-tr 4114 . . . 4  |-  ( Tr  B  <->  U. B  C_  B
)
4 unss12 3347 . . . 4  |-  ( ( U. A  C_  A  /\  U. B  C_  B
)  ->  ( U. A  u.  U. B ) 
C_  ( A  u.  B ) )
52, 3, 4syl2anb 465 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  ( U. A  u.  U. B
)  C_  ( A  u.  B ) )
61, 5syl5eqss 3222 . 2  |-  ( ( Tr  A  /\  Tr  B )  ->  U. ( A  u.  B )  C_  ( A  u.  B
) )
7 df-tr 4114 . 2  |-  ( Tr  ( A  u.  B
)  <->  U. ( A  u.  B )  C_  ( A  u.  B )
)
86, 7sylibr 203 1  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    u. cun 3150    C_ wss 3152   U.cuni 3827   Tr wtr 4113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114
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