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Theorem trin 4500
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 3608 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2 trss 4499 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
3 trss 4499 . . . . . 6  |-  ( Tr  B  ->  ( x  e.  B  ->  x  C_  B ) )
42, 3im2anan9 853 . . . . 5  |-  ( ( Tr  A  /\  Tr  B )  ->  (
( x  e.  A  /\  x  e.  B
)  ->  ( x  C_  A  /\  x  C_  B ) ) )
51, 4syl5bi 225 . . . 4  |-  ( ( Tr  A  /\  Tr  B )  ->  (
x  e.  ( A  i^i  B )  -> 
( x  C_  A  /\  x  C_  B ) ) )
6 ssin 3645 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
75, 6syl6ib 234 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  (
x  e.  ( A  i^i  B )  ->  x  C_  ( A  i^i  B ) ) )
87ralrimiv 2808 . 2  |-  ( ( Tr  A  /\  Tr  B )  ->  A. x  e.  ( A  i^i  B
) x  C_  ( A  i^i  B ) )
9 dftr3 4494 . 2  |-  ( Tr  ( A  i^i  B
)  <->  A. x  e.  ( A  i^i  B ) x  C_  ( A  i^i  B ) )
108, 9sylibr 217 1  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904   A.wral 2756    i^i cin 3389    C_ wss 3390   Tr wtr 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-in 3397  df-ss 3404  df-uni 4191  df-tr 4491
This theorem is referenced by:  ordin  5460  tcmin  8243  ingru  9258  gruina  9261  dfon2lem4  30503
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