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Theorem ordtri1 4614
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 4610 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
2 ordn2lp 4601 . . . . 5  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 imnan 412 . . . . 5  |-  ( ( A  e.  B  ->  -.  B  e.  A
)  <->  -.  ( A  e.  B  /\  B  e.  A ) )
42, 3sylibr 204 . . . 4  |-  ( Ord 
A  ->  ( A  e.  B  ->  -.  B  e.  A ) )
5 ordirr 4599 . . . . 5  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2497 . . . . . 6  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 286 . . . . 5  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibrcom 214 . . . 4  |-  ( Ord 
B  ->  ( A  =  B  ->  -.  B  e.  A ) )
94, 8jaao 496 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  ->  -.  B  e.  A
) )
10 ordtri3or 4613 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
11 df-3or 937 . . . . . 6  |-  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A ) )
1210, 11sylib 189 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  \/  B  e.  A
) )
1312orcomd 378 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  \/  ( A  e.  B  \/  A  =  B )
) )
1413ord 367 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  B  e.  A  ->  ( A  e.  B  \/  A  =  B )
) )
159, 14impbid 184 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  <->  -.  B  e.  A ) )
161, 15bitrd 245 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    C_ wss 3320   Ord word 4580
This theorem is referenced by:  ontri1  4615  ordtri2  4616  ordtri4  4618  ordtr3  4626  ordintdif  4630  ordtri2or  4677  ordsucss  4798  ordsucsssuc  4803  ordsucuniel  4804  limsssuc  4830  ssnlim  4863  smoword  6628  tfrlem15  6653  nnaword  6870  nnawordex  6880  onomeneq  7296  nndomo  7300  isfinite2  7365  unfilem1  7371  wofib  7514  cantnflem1  7645  alephgeom  7963  alephdom2  7968  cflim2  8143  fin67  8275  winainflem  8568  finminlem  26321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584
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