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Theorem treq 4027
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 3740 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
21sseq1d 3121 . . 3  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  A ) )
3 sseq2 3116 . . 3  |-  ( A  =  B  ->  ( U. B  C_  A  <->  U. B  C_  B ) )
42, 3bitrd 187 . 2  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  B ) )
5 df-tr 4022 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
6 df-tr 4022 . 2  |-  ( Tr  B  <->  U. B  C_  B
)
74, 5, 63bitr4g 222 1  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    C_ wss 3066   U.cuni 3731   Tr wtr 4021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022
This theorem is referenced by:  truni  4035  ordeq  4289  ordsucim  4411  ordom  4515  exmidonfinlem  7042
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