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

Theorem domtriord 7190
Description: Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
Assertion
Ref Expression
domtriord  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )

Proof of Theorem domtriord
StepHypRef Expression
1 sbth 7164 . . . . 5  |-  ( ( B  ~<_  A  /\  A  ~<_  B )  ->  B  ~~  A )
21expcom 425 . . . 4  |-  ( A  ~<_  B  ->  ( B  ~<_  A  ->  B  ~~  A
) )
32a1i 11 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  -> 
( B  ~<_  A  ->  B  ~~  A ) ) )
4 iman 414 . . . 4  |-  ( ( B  ~<_  A  ->  B  ~~  A )  <->  -.  ( B  ~<_  A  /\  -.  B  ~~  A ) )
5 brsdom 7067 . . . 4  |-  ( B 
~<  A  <->  ( B  ~<_  A  /\  -.  B  ~~  A ) )
64, 5xchbinxr 303 . . 3  |-  ( ( B  ~<_  A  ->  B  ~~  A )  <->  -.  B  ~<  A )
73, 6syl6ib 218 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  ->  -.  B  ~<  A ) )
8 onelss 4565 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
9 ssdomg 7090 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  C_  B  ->  A  ~<_  B ) )
108, 9syld 42 . . . . . . . . 9  |-  ( B  e.  On  ->  ( A  e.  B  ->  A  ~<_  B ) )
1110adantl 453 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  A  ~<_  B ) )
1211con3d 127 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  -.  A  e.  B ) )
13 ontri1 4557 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
1413ancoms 440 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
1512, 14sylibrd 226 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  C_  A
) )
16 ssdomg 7090 . . . . . . 7  |-  ( A  e.  On  ->  ( B  C_  A  ->  B  ~<_  A ) )
1716adantr 452 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  ->  B  ~<_  A ) )
1815, 17syld 42 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  ~<_  A ) )
19 ensym 7093 . . . . . . . 8  |-  ( B 
~~  A  ->  A  ~~  B )
20 endom 7071 . . . . . . . 8  |-  ( A 
~~  B  ->  A  ~<_  B )
2119, 20syl 16 . . . . . . 7  |-  ( B 
~~  A  ->  A  ~<_  B )
2221con3i 129 . . . . . 6  |-  ( -.  A  ~<_  B  ->  -.  B  ~~  A )
2322a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  -.  B  ~~  A ) )
2418, 23jcad 520 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  ( B  ~<_  A  /\  -.  B  ~~  A ) ) )
2524, 5syl6ibr 219 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  ~<  A ) )
2625con1d 118 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
277, 26impbid 184 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717    C_ wss 3264   class class class wbr 4154   Oncon0 4523    ~~ cen 7043    ~<_ cdom 7044    ~< csdm 7045
This theorem is referenced by:  sdomel  7191  cardsdomel  7795  alephord  7890  alephsucdom  7894  alephdom2  7902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049
  Copyright terms: Public domain W3C validator