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Theorem brdom2 6907
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 6903 . . 3  |-  ~<_  =  ( 
~<  u.  ~~  )
21eleq2i 2360 . 2  |-  ( <. A ,  B >.  e.  ~<_  <->  <. A ,  B >.  e.  (  ~<  u.  ~~  )
)
3 df-br 4040 . 2  |-  ( A  ~<_  B  <->  <. A ,  B >.  e.  ~<_  )
4 df-br 4040 . . . 4  |-  ( A 
~<  B  <->  <. A ,  B >.  e.  ~<  )
5 df-br 4040 . . . 4  |-  ( A 
~~  B  <->  <. A ,  B >.  e.  ~~  )
64, 5orbi12i 507 . . 3  |-  ( ( A  ~<  B  \/  A  ~~  B )  <->  ( <. A ,  B >.  e.  ~<  \/ 
<. A ,  B >.  e. 
~~  ) )
7 elun 3329 . . 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 1696    u. cun 3163   <.cop 3656   class class class wbr 4039    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  bren2  6908  domnsym  7003  modom  7079  carddom2  7626  axcc4dom  8083  entric  8195  entri2  8196  gchor  8265  frgpcyg  16543  iunmbl2  18930  dyadmbl  18971  ovoliunnfl  25001  ctbnfien  27004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-f1o 5278  df-en 6880  df-dom 6881  df-sdom 6882
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