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Theorem brdom2 7339
Description: Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
brdom2  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )

Proof of Theorem brdom2
StepHypRef Expression
1 dfdom2 7335 . . 3  |-  ~<_  =  ( 
~<  u.  ~~  )
21eleq2i 2507 . 2  |-  ( <. A ,  B >.  e.  ~<_  <->  <. A ,  B >.  e.  (  ~<  u.  ~~  )
)
3 df-br 4293 . 2  |-  ( A  ~<_  B  <->  <. A ,  B >.  e.  ~<_  )
4 df-br 4293 . . . 4  |-  ( A 
~<  B  <->  <. A ,  B >.  e.  ~<  )
5 df-br 4293 . . . 4  |-  ( A 
~~  B  <->  <. A ,  B >.  e.  ~~  )
64, 5orbi12i 521 . . 3  |-  ( ( A  ~<  B  \/  A  ~~  B )  <->  ( <. A ,  B >.  e.  ~<  \/ 
<. A ,  B >.  e. 
~~  ) )
7 elun 3497 . . 3  |-  ( <. A ,  B >.  e.  (  ~<  u.  ~~  )  <->  (
<. A ,  B >.  e. 
~<  \/  <. A ,  B >.  e.  ~~  ) )
86, 7bitr4i 252 . 2  |-  ( ( A  ~<  B  \/  A  ~~  B )  <->  <. A ,  B >.  e.  (  ~<  u.  ~~  ) )
92, 3, 83bitr4i 277 1  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    e. wcel 1756    u. cun 3326   <.cop 3883   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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-f1o 5425  df-en 7311  df-dom 7312  df-sdom 7313
This theorem is referenced by:  bren2  7340  domnsym  7437  modom  7513  carddom2  8147  axcc4dom  8610  entric  8721  entri2  8722  gchor  8794  frgpcyg  18006  iunmbl2  21038  dyadmbl  21080  volmeas  26647  ovoliunnfl  28433  ctbnfien  29157
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