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Theorem treq 3888
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 3617 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
21sseq1d 3000 . . 3  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  A ) )
3 sseq2 2995 . . 3  |-  ( A  =  B  ->  ( U. B  C_  A  <->  U. B  C_  B ) )
42, 3bitrd 181 . 2  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  B ) )
5 df-tr 3883 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
6 df-tr 3883 . 2  |-  ( Tr  B  <->  U. B  C_  B
)
74, 5, 63bitr4g 216 1  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    C_ wss 2945   U.cuni 3608   Tr wtr 3882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883
This theorem is referenced by:  truni  3896  ordeq  4137  ordsucim  4254  ordom  4357
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