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Theorem trin 3864
Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
Assertion
Ref Expression
trin  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )

Proof of Theorem trin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3126 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2 trss 3863 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
3 trss 3863 . . . . . 6  |-  ( Tr  B  ->  ( x  e.  B  ->  x  C_  B ) )
42, 3im2anan9 530 . . . . 5  |-  ( ( Tr  A  /\  Tr  B )  ->  (
( x  e.  A  /\  x  e.  B
)  ->  ( x  C_  A  /\  x  C_  B ) ) )
51, 4syl5bi 141 . . . 4  |-  ( ( Tr  A  /\  Tr  B )  ->  (
x  e.  ( A  i^i  B )  -> 
( x  C_  A  /\  x  C_  B ) ) )
6 ssin 3159 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
75, 6syl6ib 150 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  (
x  e.  ( A  i^i  B )  ->  x  C_  ( A  i^i  B ) ) )
87ralrimiv 2391 . 2  |-  ( ( Tr  A  /\  Tr  B )  ->  A. x  e.  ( A  i^i  B
) x  C_  ( A  i^i  B ) )
9 dftr3 3858 . 2  |-  ( Tr  ( A  i^i  B
)  <->  A. x  e.  ( A  i^i  B ) x  C_  ( A  i^i  B ) )
108, 9sylibr 137 1  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    e. wcel 1393   A.wral 2306    i^i cin 2916    C_ wss 2917   Tr wtr 3854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855
This theorem is referenced by:  ordin  4122
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