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Theorem ordtri1 4425
Description: A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )

Proof of Theorem ordtri1
StepHypRef Expression
1 ordsseleq 4421 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
2 ordn2lp 4412 . . . . 5  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 imnan 411 . . . . 5  |-  ( ( A  e.  B  ->  -.  B  e.  A
)  <->  -.  ( A  e.  B  /\  B  e.  A ) )
42, 3sylibr 203 . . . 4  |-  ( Ord 
A  ->  ( A  e.  B  ->  -.  B  e.  A ) )
5 ordirr 4410 . . . . 5  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2344 . . . . . 6  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 285 . . . . 5  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibrcom 213 . . . 4  |-  ( Ord 
B  ->  ( A  =  B  ->  -.  B  e.  A ) )
94, 8jaao 495 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  ->  -.  B  e.  A
) )
10 ordtri3or 4424 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
11 df-3or 935 . . . . . 6  |-  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A ) )
1210, 11sylib 188 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  \/  B  e.  A
) )
1312orcomd 377 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  \/  ( A  e.  B  \/  A  =  B )
) )
1413ord 366 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  B  e.  A  ->  ( A  e.  B  \/  A  =  B )
) )
159, 14impbid 183 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  <->  -.  B  e.  A ) )
161, 15bitrd 244 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684    C_ wss 3152   Ord word 4391
This theorem is referenced by:  ontri1  4426  ordtri2  4427  ordtri4  4429  ordtr3  4437  ordintdif  4441  ordtri2or  4488  ordsucss  4609  ordsucsssuc  4614  ordsucuniel  4615  limsssuc  4641  ssnlim  4674  smoword  6383  tfrlem15  6408  nnaword  6625  nnawordex  6635  onomeneq  7050  nndomo  7054  isfinite2  7115  unfilem1  7121  wofib  7260  cantnflem1  7391  alephgeom  7709  alephdom2  7714  cflim2  7889  fin67  8021  winainflem  8315  finminlem  26231
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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