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Theorem ordtri1 5418
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 5414 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
2 ordn2lp 5405 . . . . 5  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 imnan 423 . . . . 5  |-  ( ( A  e.  B  ->  -.  B  e.  A
)  <->  -.  ( A  e.  B  /\  B  e.  A ) )
42, 3sylibr 215 . . . 4  |-  ( Ord 
A  ->  ( A  e.  B  ->  -.  B  e.  A ) )
5 ordirr 5403 . . . . 5  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2495 . . . . . 6  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 295 . . . . 5  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibrcom 225 . . . 4  |-  ( Ord 
B  ->  ( A  =  B  ->  -.  B  e.  A ) )
94, 8jaao 511 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  ->  -.  B  e.  A
) )
10 ordtri3or 5417 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
11 df-3or 983 . . . . . 6  |-  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A ) )
1210, 11sylib 199 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  \/  B  e.  A
) )
1312orcomd 389 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  \/  ( A  e.  B  \/  A  =  B )
) )
1413ord 378 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  B  e.  A  ->  ( A  e.  B  \/  A  =  B )
) )
159, 14impbid 193 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  <->  -.  B  e.  A ) )
161, 15bitrd 256 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1872    C_ wss 3379   Ord word 5384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-tr 4462  df-eprel 4707  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-ord 5388
This theorem is referenced by:  ontri1  5419  ordtri2  5420  ordtri4  5422  ordtr3  5430  ordintdif  5434  ordtri2or  5480  ordsucss  6603  ordsucsssuc  6608  ordsucuniel  6609  limsssuc  6635  ssnlim  6668  smoword  7040  tfrlem15  7065  nnaword  7283  nnawordex  7293  onomeneq  7715  nndomo  7719  isfinite2  7782  unfilem1  7788  wofib  8013  cantnflem1  8146  alephgeom  8464  alephdom2  8469  cflim2  8644  fin67  8776  winainflem  9069  finminlem  30923
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