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Theorem domtriord 7023
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 6997 . . . . 5  |-  ( ( B  ~<_  A  /\  A  ~<_  B )  ->  B  ~~  A )
21expcom 424 . . . 4  |-  ( A  ~<_  B  ->  ( B  ~<_  A  ->  B  ~~  A
) )
32a1i 10 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  -> 
( B  ~<_  A  ->  B  ~~  A ) ) )
4 iman 413 . . . 4  |-  ( ( B  ~<_  A  ->  B  ~~  A )  <->  -.  ( B  ~<_  A  /\  -.  B  ~~  A ) )
5 brsdom 6900 . . . 4  |-  ( B 
~<  A  <->  ( B  ~<_  A  /\  -.  B  ~~  A ) )
64, 5xchbinxr 302 . . 3  |-  ( ( B  ~<_  A  ->  B  ~~  A )  <->  -.  B  ~<  A )
73, 6syl6ib 217 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  ->  -.  B  ~<  A ) )
8 onelss 4450 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
9 ssdomg 6923 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  C_  B  ->  A  ~<_  B ) )
108, 9syld 40 . . . . . . . . 9  |-  ( B  e.  On  ->  ( A  e.  B  ->  A  ~<_  B ) )
1110adantl 452 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  A  ~<_  B ) )
1211con3d 125 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  -.  A  e.  B ) )
13 ontri1 4442 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
1413ancoms 439 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
1512, 14sylibrd 225 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  C_  A
) )
16 ssdomg 6923 . . . . . . 7  |-  ( A  e.  On  ->  ( B  C_  A  ->  B  ~<_  A ) )
1716adantr 451 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  ->  B  ~<_  A ) )
1815, 17syld 40 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  ~<_  A ) )
19 ensym 6926 . . . . . . . 8  |-  ( B 
~~  A  ->  A  ~~  B )
20 endom 6904 . . . . . . . 8  |-  ( A 
~~  B  ->  A  ~<_  B )
2119, 20syl 15 . . . . . . 7  |-  ( B 
~~  A  ->  A  ~<_  B )
2221con3i 127 . . . . . 6  |-  ( -.  A  ~<_  B  ->  -.  B  ~~  A )
2322a1i 10 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  -.  B  ~~  A ) )
2418, 23jcad 519 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  ( B  ~<_  A  /\  -.  B  ~~  A ) ) )
2524, 5syl6ibr 218 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  ~<  A ) )
2625con1d 116 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
277, 26impbid 183 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696    C_ wss 3165   class class class wbr 4039   Oncon0 4408    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  sdomel  7024  cardsdomel  7623  alephord  7718  alephsucdom  7722  alephdom2  7730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
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