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Theorem domtriord 7557
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 7531 . . . . 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 7432 . . . 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 4859 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
9 ssdomg 7455 . . . . . . . . . 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 4851 . . . . . . . 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 7455 . . . . . . 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 7458 . . . . . . . 8  |-  ( B 
~~  A  ->  A  ~~  B )
20 endom 7436 . . . . . . . 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 1758    C_ wss 3426   class class class wbr 4390   Oncon0 4817    ~~ cen 7407    ~<_ cdom 7408    ~< csdm 7409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413
This theorem is referenced by:  sdomel  7558  cardsdomel  8245  alephord  8346  alephsucdom  8350  alephdom2  8358
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