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Theorem ordtri1 4897
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 4893 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
2 ordn2lp 4884 . . . . 5  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 imnan 422 . . . . 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 4882 . . . . 5  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2514 . . . . . 6  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 294 . . . . 5  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibrcom 222 . . . 4  |-  ( Ord 
B  ->  ( A  =  B  ->  -.  B  e.  A ) )
94, 8jaao 509 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  ->  -.  B  e.  A
) )
10 ordtri3or 4896 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
11 df-3or 973 . . . . . 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 388 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  \/  ( A  e.  B  \/  A  =  B )
) )
1413ord 377 . . 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 368    /\ wa 369    \/ w3o 971    = wceq 1381    e. wcel 1802    C_ wss 3458   Ord word 4863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-tr 4527  df-eprel 4777  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867
This theorem is referenced by:  ontri1  4898  ordtri2  4899  ordtri4  4901  ordtr3  4909  ordintdif  4913  ordtri2or  4959  ordsucss  6634  ordsucsssuc  6639  ordsucuniel  6640  limsssuc  6666  ssnlim  6699  smoword  7035  tfrlem15  7059  nnaword  7274  nnawordex  7284  onomeneq  7705  nndomo  7709  isfinite2  7776  unfilem1  7782  wofib  7968  cantnflem1  8106  cantnflem1OLD  8129  alephgeom  8461  alephdom2  8466  cflim2  8641  fin67  8773  winainflem  9069  finminlem  30104
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