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Theorem brsdom 6927
Description: Strict dominance relation, meaning " B is strictly greater in size than  A." Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)
Assertion
Ref Expression
brsdom  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )

Proof of Theorem brsdom
StepHypRef Expression
1 df-sdom 6909 . . 3  |-  ~<  =  (  ~<_  \  ~~  )
21eleq2i 2380 . 2  |-  ( <. A ,  B >.  e. 
~< 
<-> 
<. A ,  B >.  e.  (  ~<_  \  ~~  ) )
3 df-br 4061 . 2  |-  ( A 
~<  B  <->  <. A ,  B >.  e.  ~<  )
4 df-br 4061 . . . 4  |-  ( A  ~<_  B  <->  <. A ,  B >.  e.  ~<_  )
5 df-br 4061 . . . . 5  |-  ( A 
~~  B  <->  <. A ,  B >.  e.  ~~  )
65notbii 287 . . . 4  |-  ( -.  A  ~~  B  <->  -.  <. A ,  B >.  e.  ~~  )
74, 6anbi12i 678 . . 3  |-  ( ( A  ~<_  B  /\  -.  A  ~~  B )  <->  ( <. A ,  B >.  e.  ~<_  /\  -.  <. A ,  B >.  e. 
~~  ) )
8 eldif 3196 . . 3  |-  ( <. A ,  B >.  e.  (  ~<_  \  ~~  )  <->  ( <. A ,  B >.  e.  ~<_  /\  -.  <. A ,  B >.  e. 
~~  ) )
97, 8bitr4i 243 . 2  |-  ( ( A  ~<_  B  /\  -.  A  ~~  B )  <->  <. A ,  B >.  e.  (  ~<_  \  ~~  ) )
102, 3, 93bitr4i 268 1  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    e. wcel 1701    \ cdif 3183   <.cop 3677   class class class wbr 4060    ~~ cen 6903    ~<_ cdom 6904    ~< csdm 6905
This theorem is referenced by:  sdomdom  6932  sdomnen  6933  0sdomg  7033  sdomdomtr  7037  domsdomtr  7039  domtriord  7050  canth2  7057  php2  7089  php3  7090  nnsdomo  7098  nnsdomg  7161  card2inf  7314  cardsdomelir  7651  cardsdom2  7666  fidomtri2  7672  cardmin2  7676  alephordi  7746  alephord  7747  isfin4-3  7986  isfin5-2  8062  canthnum  8316  canthwe  8318  canthp1  8321  gchcdaidm  8335  gchxpidm  8336  gchhar  8338  axgroth6  8495  hashsdom  11410  ruc  12568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-dif 3189  df-br 4061  df-sdom 6909
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