ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  trin Unicode version

Theorem trin 3887
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 3156 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2 trss 3886 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
3 trss 3886 . . . . . 6  |-  ( Tr  B  ->  ( x  e.  B  ->  x  C_  B ) )
42, 3im2anan9 563 . . . . 5  |-  ( ( Tr  A  /\  Tr  B )  ->  (
( x  e.  A  /\  x  e.  B
)  ->  ( x  C_  A  /\  x  C_  B ) ) )
51, 4syl5bi 150 . . . 4  |-  ( ( Tr  A  /\  Tr  B )  ->  (
x  e.  ( A  i^i  B )  -> 
( x  C_  A  /\  x  C_  B ) ) )
6 ssin 3189 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
75, 6syl6ib 159 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  (
x  e.  ( A  i^i  B )  ->  x  C_  ( A  i^i  B ) ) )
87ralrimiv 2434 . 2  |-  ( ( Tr  A  /\  Tr  B )  ->  A. x  e.  ( A  i^i  B
) x  C_  ( A  i^i  B ) )
9 dftr3 3881 . 2  |-  ( Tr  ( A  i^i  B
)  <->  A. x  e.  ( A  i^i  B ) x  C_  ( A  i^i  B ) )
108, 9sylibr 132 1  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   A.wral 2349    i^i cin 2973    C_ wss 2974   Tr wtr 3877
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-uni 3604  df-tr 3878
This theorem is referenced by:  ordin  4142
  Copyright terms: Public domain W3C validator