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

Theorem treq 4102
Description: Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
Assertion
Ref Expression
treq  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)

Proof of Theorem treq
StepHypRef Expression
1 unieq 3814 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
21sseq1d 3182 . . 3  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  A ) )
3 sseq2 3177 . . 3  |-  ( A  =  B  ->  ( U. B  C_  A  <->  U. B  C_  B ) )
42, 3bitrd 188 . 2  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  B ) )
5 df-tr 4097 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
6 df-tr 4097 . 2  |-  ( Tr  B  <->  U. B  C_  B
)
74, 5, 63bitr4g 223 1  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    C_ wss 3127   U.cuni 3805   Tr wtr 4096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-in 3133  df-ss 3140  df-uni 3806  df-tr 4097
This theorem is referenced by:  truni  4110  ordeq  4366  ordsucim  4493  ordom  4600  exmidonfinlem  7182
  Copyright terms: Public domain W3C validator