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Theorem ordeq 4399
Description: Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
Assertion
Ref Expression
ordeq  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )

Proof of Theorem ordeq
StepHypRef Expression
1 treq 4119 . . 3  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
2 weeq2 4382 . . 3  |-  ( A  =  B  ->  (  _E  We  A  <->  _E  We  B ) )
31, 2anbi12d 691 . 2  |-  ( A  =  B  ->  (
( Tr  A  /\  _E  We  A )  <->  ( Tr  B  /\  _E  We  B
) ) )
4 df-ord 4395 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
5 df-ord 4395 . 2  |-  ( Ord 
B  <->  ( Tr  B  /\  _E  We  B ) )
63, 4, 53bitr4g 279 1  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   Tr wtr 4113    _E cep 4303    We wwe 4351   Ord word 4391
This theorem is referenced by:  elong  4400  limeq  4404  ordelord  4414  ordun  4494  ordeleqon  4580  ordsuc  4605  ordzsl  4636  issmo  6365  issmo2  6366  smoeq  6367  smores  6369  smores2  6371  smodm2  6372  smoiso  6379  tfrlem8  6400  ordtypelem5  7237  ordtypelem7  7239  oicl  7244  oieu  7254
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-ral 2548  df-rex 2549  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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