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Theorem ordtri1 4900
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 4896 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
2 ordn2lp 4887 . . . . 5  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 imnan 420 . . . . 5  |-  ( ( A  e.  B  ->  -.  B  e.  A
)  <->  -.  ( A  e.  B  /\  B  e.  A ) )
42, 3sylibr 212 . . . 4  |-  ( Ord 
A  ->  ( A  e.  B  ->  -.  B  e.  A ) )
5 ordirr 4885 . . . . 5  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2527 . . . . . 6  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 292 . . . . 5  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibrcom 222 . . . 4  |-  ( Ord 
B  ->  ( A  =  B  ->  -.  B  e.  A ) )
94, 8jaao 507 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  ->  -.  B  e.  A
) )
10 ordtri3or 4899 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
11 df-3or 972 . . . . . 6  |-  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A ) )
1210, 11sylib 196 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  \/  B  e.  A
) )
1312orcomd 386 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  \/  ( A  e.  B  \/  A  =  B )
) )
1413ord 375 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  B  e.  A  ->  ( A  e.  B  \/  A  =  B )
) )
159, 14impbid 191 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  <->  -.  B  e.  A ) )
161, 15bitrd 253 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    \/ w3o 970    = wceq 1398    e. wcel 1823    C_ wss 3461   Ord word 4866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870
This theorem is referenced by:  ontri1  4901  ordtri2  4902  ordtri4  4904  ordtr3  4912  ordintdif  4916  ordtri2or  4962  ordsucss  6626  ordsucsssuc  6631  ordsucuniel  6632  limsssuc  6658  ssnlim  6691  smoword  7029  tfrlem15  7053  nnaword  7268  nnawordex  7278  onomeneq  7700  nndomo  7704  isfinite2  7770  unfilem1  7776  wofib  7962  cantnflem1  8099  cantnflem1OLD  8122  alephgeom  8454  alephdom2  8459  cflim2  8634  fin67  8766  winainflem  9060  finminlem  30376
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