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Theorem treq 4119
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 3836 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
21sseq1d 3205 . . 3  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  A ) )
3 sseq2 3200 . . 3  |-  ( A  =  B  ->  ( U. B  C_  A  <->  U. B  C_  B ) )
42, 3bitrd 244 . 2  |-  ( A  =  B  ->  ( U. A  C_  A  <->  U. B  C_  B ) )
5 df-tr 4114 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
6 df-tr 4114 . 2  |-  ( Tr  B  <->  U. B  C_  B
)
74, 5, 63bitr4g 279 1  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    C_ wss 3152   U.cuni 3827   Tr wtr 4113
This theorem is referenced by:  truni  4127  ordeq  4399  trsuc2OLD  4477  trcl  7410  tz9.1  7411  tz9.1c  7412  tctr  7425  tcmin  7426  tc2  7427  r1tr  7448  r1elssi  7477  tcrank  7554  iswun  8326  tskr1om2  8390  elgrug  8414  grutsk  8444  dfon2lem1  24139  dfon2lem3  24141  dfon2lem4  24142  dfon2lem5  24143  dfon2lem6  24144  dfon2lem7  24145  dfon2lem8  24146  dfon2  24148  dford3lem1  27119  dford3lem2  27120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114
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