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Theorem entric 8421
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 8420 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
21biimprd 215 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
3 brdom2 7128 . . . . 5  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
42, 3syl6ib 218 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  B  ~<  A  ->  ( A  ~<  B  \/  A  ~~  B
) ) )
54con1d 118 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ( A 
~<  B  \/  A  ~~  B )  ->  B  ~<  A ) )
65orrd 368 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ~<  B  \/  A  ~~  B
)  \/  B  ~<  A ) )
7 df-3or 937 . 2  |-  ( ( A  ~<  B  \/  A  ~~  B  \/  B  ~<  A )  <->  ( ( A  ~<  B  \/  A  ~~  B )  \/  B  ~<  A ) )
86, 7sylibr 204 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<  B  \/  A  ~~  B  \/  B  ~<  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    e. wcel 1725   class class class wbr 4204    ~~ cen 7097    ~<_ cdom 7098    ~< csdm 7099
This theorem is referenced by:  entri2  8422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-ac2 8332
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-riota 6540  df-recs 6624  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-card 7815  df-ac 7986
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