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Theorem brdom2 7129
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 7125 . . 3  |-  ~<_  =  ( 
~<  u.  ~~  )
21eleq2i 2499 . 2  |-  ( <. A ,  B >.  e.  ~<_  <->  <. A ,  B >.  e.  (  ~<  u.  ~~  )
)
3 df-br 4205 . 2  |-  ( A  ~<_  B  <->  <. A ,  B >.  e.  ~<_  )
4 df-br 4205 . . . 4  |-  ( A 
~<  B  <->  <. A ,  B >.  e.  ~<  )
5 df-br 4205 . . . 4  |-  ( A 
~~  B  <->  <. A ,  B >.  e.  ~~  )
64, 5orbi12i 508 . . 3  |-  ( ( A  ~<  B  \/  A  ~~  B )  <->  ( <. A ,  B >.  e.  ~<  \/ 
<. A ,  B >.  e. 
~~  ) )
7 elun 3480 . . 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 1725    u. cun 3310   <.cop 3809   class class class wbr 4204    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100
This theorem is referenced by:  bren2  7130  domnsym  7225  modom  7301  carddom2  7856  axcc4dom  8313  entric  8424  entri2  8425  gchor  8494  frgpcyg  16846  iunmbl2  19443  dyadmbl  19484  volmeas  24579  ovoliunnfl  26238  ctbnfien  26870
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-f1o 5453  df-en 7102  df-dom 7103  df-sdom 7104
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