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

Theorem ordtri3 4909
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 4891 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq2 2535 . . . . . . 7  |-  ( A  =  B  ->  ( A  e.  A  <->  A  e.  B ) )
32notbid 294 . . . . . 6  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  A  e.  B ) )
41, 3syl5ib 219 . . . . 5  |-  ( A  =  B  ->  ( Ord  A  ->  -.  A  e.  B ) )
5 ordirr 4891 . . . . . 6  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2535 . . . . . . 7  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 294 . . . . . 6  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibr 221 . . . . 5  |-  ( A  =  B  ->  ( Ord  B  ->  -.  B  e.  A ) )
94, 8anim12d 563 . . . 4  |-  ( A  =  B  ->  (
( Ord  A  /\  Ord  B )  ->  ( -.  A  e.  B  /\  -.  B  e.  A
) ) )
109com12 31 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  ->  ( -.  A  e.  B  /\  -.  B  e.  A
) ) )
11 pm4.56 495 . . 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 4905 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
14 df-3or 969 . . . . 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 527 . . . 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 377 . 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 368    /\ wa 369    \/ w3o 967    = wceq 1374    e. wcel 1762   Ord word 4872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-tr 4536  df-eprel 4786  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876
This theorem is referenced by:  ordunisuc2  6652  tz7.48lem  7098  oacan  7189  omcan  7210  oecan  7230  omsmo  7295  omopthi  7298  inf3lem6  8041  cantnfp1lem3  8090  cantnfp1lem3OLD  8116  infpssrlem5  8678  fin23lem24  8693  isf32lem4  8727  om2uzf1oi  12022  nodenselem4  29009
  Copyright terms: Public domain W3C validator