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Theorem domtriord 7718
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 7692 . . . . 5  |-  ( ( B  ~<_  A  /\  A  ~<_  B )  ->  B  ~~  A )
21expcom 437 . . . 4  |-  ( A  ~<_  B  ->  ( B  ~<_  A  ->  B  ~~  A
) )
32a1i 11 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  -> 
( B  ~<_  A  ->  B  ~~  A ) ) )
4 iman 426 . . . 4  |-  ( ( B  ~<_  A  ->  B  ~~  A )  <->  -.  ( B  ~<_  A  /\  -.  B  ~~  A ) )
5 brsdom 7592 . . . 4  |-  ( B 
~<  A  <->  ( B  ~<_  A  /\  -.  B  ~~  A ) )
64, 5xchbinxr 313 . . 3  |-  ( ( B  ~<_  A  ->  B  ~~  A )  <->  -.  B  ~<  A )
73, 6syl6ib 230 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  ->  -.  B  ~<  A ) )
8 onelss 5465 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
9 ssdomg 7615 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  C_  B  ->  A  ~<_  B ) )
108, 9syld 45 . . . . . . . . 9  |-  ( B  e.  On  ->  ( A  e.  B  ->  A  ~<_  B ) )
1110adantl 468 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  A  ~<_  B ) )
1211con3d 139 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  -.  A  e.  B ) )
13 ontri1 5457 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
1413ancoms 455 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
1512, 14sylibrd 238 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  C_  A
) )
16 ssdomg 7615 . . . . . . 7  |-  ( A  e.  On  ->  ( B  C_  A  ->  B  ~<_  A ) )
1716adantr 467 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  ->  B  ~<_  A ) )
1815, 17syld 45 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  ~<_  A ) )
19 ensym 7618 . . . . . . . 8  |-  ( B 
~~  A  ->  A  ~~  B )
20 endom 7596 . . . . . . . 8  |-  ( A 
~~  B  ->  A  ~<_  B )
2119, 20syl 17 . . . . . . 7  |-  ( B 
~~  A  ->  A  ~<_  B )
2221con3i 141 . . . . . 6  |-  ( -.  A  ~<_  B  ->  -.  B  ~~  A )
2322a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  -.  B  ~~  A ) )
2418, 23jcad 536 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  ( B  ~<_  A  /\  -.  B  ~~  A ) ) )
2524, 5syl6ibr 231 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  ~<_  B  ->  B  ~<  A ) )
2625con1d 128 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
277, 26impbid 194 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    e. wcel 1887    C_ wss 3404   class class class wbr 4402   Oncon0 5423    ~~ cen 7566    ~<_ cdom 7567    ~< csdm 7568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572
This theorem is referenced by:  sdomel  7719  cardsdomel  8408  alephord  8506  alephsucdom  8510  alephdom2  8518
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