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Theorem entric 8721
Description: Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.)
Assertion
Ref Expression
entric  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<  B  \/  A  ~~  B  \/  B  ~<  A ) )

Proof of Theorem entric
StepHypRef Expression
1 domtri 8720 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
21biimprd 223 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
3 brdom2 7339 . . . . 5  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
42, 3syl6ib 226 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  B  ~<  A  ->  ( A  ~<  B  \/  A  ~~  B
) ) )
54con1d 124 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ( A 
~<  B  \/  A  ~~  B )  ->  B  ~<  A ) )
65orrd 378 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ~<  B  \/  A  ~~  B
)  \/  B  ~<  A ) )
7 df-3or 966 . 2  |-  ( ( A  ~<  B  \/  A  ~~  B  \/  B  ~<  A )  <->  ( ( A  ~<  B  \/  A  ~~  B )  \/  B  ~<  A ) )
86, 7sylibr 212 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<  B  \/  A  ~~  B  \/  B  ~<  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 964    e. wcel 1756   class class class wbr 4292    ~~ cen 7307    ~<_ cdom 7308    ~< csdm 7309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-ac2 8632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-recs 6832  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-card 8109  df-ac 8286
This theorem is referenced by:  entri2  8722
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