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Theorem brdom2 6891
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 6887 . . 3  |-  ~<_  =  ( 
~<  u.  ~~  )
21eleq2i 2347 . 2  |-  ( <. A ,  B >.  e.  ~<_  <->  <. A ,  B >.  e.  (  ~<  u.  ~~  )
)
3 df-br 4024 . 2  |-  ( A  ~<_  B  <->  <. A ,  B >.  e.  ~<_  )
4 df-br 4024 . . . 4  |-  ( A 
~<  B  <->  <. A ,  B >.  e.  ~<  )
5 df-br 4024 . . . 4  |-  ( A 
~~  B  <->  <. A ,  B >.  e.  ~~  )
64, 5orbi12i 507 . . 3  |-  ( ( A  ~<  B  \/  A  ~~  B )  <->  ( <. A ,  B >.  e.  ~<  \/ 
<. A ,  B >.  e. 
~~  ) )
7 elun 3316 . . 3  |-  ( <. A ,  B >.  e.  (  ~<  u.  ~~  )  <->  (
<. A ,  B >.  e. 
~<  \/  <. A ,  B >.  e.  ~~  ) )
86, 7bitr4i 243 . 2  |-  ( ( A  ~<  B  \/  A  ~~  B )  <->  <. A ,  B >.  e.  (  ~<  u.  ~~  ) )
92, 3, 83bitr4i 268 1  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    e. wcel 1684    u. cun 3150   <.cop 3643   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  bren2  6892  domnsym  6987  modom  7063  carddom2  7610  axcc4dom  8067  entric  8179  entri2  8180  gchor  8249  frgpcyg  16527  iunmbl2  18914  dyadmbl  18955  ctbnfien  26901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-f1o 5262  df-en 6864  df-dom 6865  df-sdom 6866
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