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Theorem domtriord 7660
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 7634 . . . . 5  |-  ( ( B  ~<_  A  /\  A  ~<_  B )  ->  B  ~~  A )
21expcom 435 . . . 4  |-  ( A  ~<_  B  ->  ( B  ~<_  A  ->  B  ~~  A
) )
32a1i 11 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  -> 
( B  ~<_  A  ->  B  ~~  A ) ) )
4 iman 424 . . . 4  |-  ( ( B  ~<_  A  ->  B  ~~  A )  <->  -.  ( B  ~<_  A  /\  -.  B  ~~  A ) )
5 brsdom 7535 . . . 4  |-  ( B 
~<  A  <->  ( B  ~<_  A  /\  -.  B  ~~  A ) )
64, 5xchbinxr 311 . . 3  |-  ( ( B  ~<_  A  ->  B  ~~  A )  <->  -.  B  ~<  A )
73, 6syl6ib 226 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  ->  -.  B  ~<  A ) )
8 onelss 4920 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
9 ssdomg 7558 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  C_  B  ->  A  ~<_  B ) )
108, 9syld 44 . . . . . . . . 9  |-  ( B  e.  On  ->  ( A  e.  B  ->  A  ~<_  B ) )
1110adantl 466 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  A  ~<_  B ) )
1211con3d 133 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  -.  A  e.  B ) )
13 ontri1 4912 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
1413ancoms 453 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
1512, 14sylibrd 234 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  C_  A
) )
16 ssdomg 7558 . . . . . . 7  |-  ( A  e.  On  ->  ( B  C_  A  ->  B  ~<_  A ) )
1716adantr 465 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  ->  B  ~<_  A ) )
1815, 17syld 44 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  ~<_  A ) )
19 ensym 7561 . . . . . . . 8  |-  ( B 
~~  A  ->  A  ~~  B )
20 endom 7539 . . . . . . . 8  |-  ( A 
~~  B  ->  A  ~<_  B )
2119, 20syl 16 . . . . . . 7  |-  ( B 
~~  A  ->  A  ~<_  B )
2221con3i 135 . . . . . 6  |-  ( -.  A  ~<_  B  ->  -.  B  ~~  A )
2322a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  -.  B  ~~  A ) )
2418, 23jcad 533 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  ( B  ~<_  A  /\  -.  B  ~~  A ) ) )
2524, 5syl6ibr 227 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  ~<  A ) )
2625con1d 124 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
277, 26impbid 191 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    C_ wss 3476   class class class wbr 4447   Oncon0 4878    ~~ cen 7510    ~<_ cdom 7511    ~< csdm 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516
This theorem is referenced by:  sdomel  7661  cardsdomel  8351  alephord  8452  alephsucdom  8456  alephdom2  8464
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