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Theorem treq 4122
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 3833 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
21sseq1d 3199 . . 3  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  A ) )
3 sseq2 3194 . . 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 4117 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
6 df-tr 4117 . 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 1364    C_ wss 3144   U.cuni 3824   Tr wtr 4116
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117
This theorem is referenced by:  truni  4130  ordeq  4390  ordsucim  4517  ordom  4624  exmidonfinlem  7221
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