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Theorem brdom2 7075
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 7071 . . 3  |-  ~<_  =  ( 
~<  u.  ~~  )
21eleq2i 2453 . 2  |-  ( <. A ,  B >.  e.  ~<_  <->  <. A ,  B >.  e.  (  ~<  u.  ~~  )
)
3 df-br 4156 . 2  |-  ( A  ~<_  B  <->  <. A ,  B >.  e.  ~<_  )
4 df-br 4156 . . . 4  |-  ( A 
~<  B  <->  <. A ,  B >.  e.  ~<  )
5 df-br 4156 . . . 4  |-  ( A 
~~  B  <->  <. A ,  B >.  e.  ~~  )
64, 5orbi12i 508 . . 3  |-  ( ( A  ~<  B  \/  A  ~~  B )  <->  ( <. A ,  B >.  e.  ~<  \/ 
<. A ,  B >.  e. 
~~  ) )
7 elun 3433 . . 3  |-  ( <. A ,  B >.  e.  (  ~<  u.  ~~  )  <->  (
<. A ,  B >.  e. 
~<  \/  <. A ,  B >.  e.  ~~  ) )
86, 7bitr4i 244 . 2  |-  ( ( A  ~<  B  \/  A  ~~  B )  <->  <. A ,  B >.  e.  (  ~<  u.  ~~  ) )
92, 3, 83bitr4i 269 1  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    e. wcel 1717    u. cun 3263   <.cop 3762   class class class wbr 4155    ~~ cen 7044    ~<_ cdom 7045    ~< csdm 7046
This theorem is referenced by:  bren2  7076  domnsym  7171  modom  7247  carddom2  7799  axcc4dom  8256  entric  8367  entri2  8368  gchor  8437  frgpcyg  16779  iunmbl2  19320  dyadmbl  19361  volmeas  24383  ovoliunnfl  25955  ctbnfien  26572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-f1o 5403  df-en 7048  df-dom 7049  df-sdom 7050
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