MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtri3 Structured version   Unicode version

Theorem ordtri3 5445
Description: A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )

Proof of Theorem ordtri3
StepHypRef Expression
1 ordirr 5427 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq2 2475 . . . . . . 7  |-  ( A  =  B  ->  ( A  e.  A  <->  A  e.  B ) )
32notbid 292 . . . . . 6  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  A  e.  B ) )
41, 3syl5ib 219 . . . . 5  |-  ( A  =  B  ->  ( Ord  A  ->  -.  A  e.  B ) )
5 ordirr 5427 . . . . . 6  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2475 . . . . . . 7  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 292 . . . . . 6  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibr 221 . . . . 5  |-  ( A  =  B  ->  ( Ord  B  ->  -.  B  e.  A ) )
94, 8anim12d 561 . . . 4  |-  ( A  =  B  ->  (
( Ord  A  /\  Ord  B )  ->  ( -.  A  e.  B  /\  -.  B  e.  A
) ) )
109com12 29 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  ->  ( -.  A  e.  B  /\  -.  B  e.  A
) ) )
11 pm4.56 493 . . 3  |-  ( ( -.  A  e.  B  /\  -.  B  e.  A
)  <->  -.  ( A  e.  B  \/  B  e.  A ) )
1210, 11syl6ib 226 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  ->  -.  ( A  e.  B  \/  B  e.  A )
) )
13 ordtri3or 5441 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
14 df-3or 975 . . . . 5  |-  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A ) )
1513, 14sylib 196 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  \/  B  e.  A
) )
16 or32 525 . . . 4  |-  ( ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A )  <->  ( ( A  e.  B  \/  B  e.  A )  \/  A  =  B
) )
1715, 16sylib 196 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  B  e.  A )  \/  A  =  B
) )
1817ord 375 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  e.  B  \/  B  e.  A
)  ->  A  =  B ) )
1912, 18impbid 191 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    \/ w3o 973    = wceq 1405    e. wcel 1842   Ord word 5408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 5412
This theorem is referenced by:  ordunisuc2  6661  tz7.48lem  7142  oacan  7233  omcan  7254  oecan  7274  omsmo  7339  omopthi  7342  inf3lem6  8082  cantnfp1lem3  8130  cantnfp1lem3OLD  8156  infpssrlem5  8718  fin23lem24  8733  isf32lem4  8767  om2uzf1oi  12103  nodenselem4  30131
  Copyright terms: Public domain W3C validator