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Theorem trin 4304
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 3522 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2 trss 4303 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
3 trss 4303 . . . . . 6  |-  ( Tr  B  ->  ( x  e.  B  ->  x  C_  B ) )
42, 3im2anan9 809 . . . . 5  |-  ( ( Tr  A  /\  Tr  B )  ->  (
( x  e.  A  /\  x  e.  B
)  ->  ( x  C_  A  /\  x  C_  B ) ) )
51, 4syl5bi 209 . . . 4  |-  ( ( Tr  A  /\  Tr  B )  ->  (
x  e.  ( A  i^i  B )  -> 
( x  C_  A  /\  x  C_  B ) ) )
6 ssin 3555 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
75, 6syl6ib 218 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  (
x  e.  ( A  i^i  B )  ->  x  C_  ( A  i^i  B ) ) )
87ralrimiv 2780 . 2  |-  ( ( Tr  A  /\  Tr  B )  ->  A. x  e.  ( A  i^i  B
) x  C_  ( A  i^i  B ) )
9 dftr3 4298 . 2  |-  ( Tr  ( A  i^i  B
)  <->  A. x  e.  ( A  i^i  B ) x  C_  ( A  i^i  B ) )
108, 9sylibr 204 1  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697    i^i cin 3311    C_ wss 3312   Tr wtr 4294
This theorem is referenced by:  ordin  4603  tcmin  7672  ingru  8682  gruina  8685  dfon2lem4  25405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295
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