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Theorem brsdom 7122
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 7104 . . 3  |-  ~<  =  (  ~<_  \  ~~  )
21eleq2i 2499 . 2  |-  ( <. A ,  B >.  e. 
~< 
<-> 
<. A ,  B >.  e.  (  ~<_  \  ~~  ) )
3 df-br 4205 . 2  |-  ( A 
~<  B  <->  <. A ,  B >.  e.  ~<  )
4 df-br 4205 . . . 4  |-  ( A  ~<_  B  <->  <. A ,  B >.  e.  ~<_  )
5 df-br 4205 . . . . 5  |-  ( A 
~~  B  <->  <. A ,  B >.  e.  ~~  )
65notbii 288 . . . 4  |-  ( -.  A  ~~  B  <->  -.  <. A ,  B >.  e.  ~~  )
74, 6anbi12i 679 . . 3  |-  ( ( A  ~<_  B  /\  -.  A  ~~  B )  <->  ( <. A ,  B >.  e.  ~<_  /\  -.  <. A ,  B >.  e. 
~~  ) )
8 eldif 3322 . . 3  |-  ( <. A ,  B >.  e.  (  ~<_  \  ~~  )  <->  ( <. A ,  B >.  e.  ~<_  /\  -.  <. A ,  B >.  e. 
~~  ) )
97, 8bitr4i 244 . 2  |-  ( ( A  ~<_  B  /\  -.  A  ~~  B )  <->  <. A ,  B >.  e.  (  ~<_  \  ~~  ) )
102, 3, 93bitr4i 269 1  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    e. wcel 1725    \ cdif 3309   <.cop 3809   class class class wbr 4204    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100
This theorem is referenced by:  sdomdom  7127  sdomnen  7128  0sdomg  7228  sdomdomtr  7232  domsdomtr  7234  domtriord  7245  canth2  7252  php2  7284  php3  7285  nnsdomo  7293  nnsdomg  7358  card2inf  7515  cardsdomelir  7852  cardsdom2  7867  fidomtri2  7873  cardmin2  7877  alephordi  7947  alephord  7948  isfin4-3  8187  isfin5-2  8263  canthnum  8516  canthwe  8518  canthp1  8521  gchcdaidm  8535  gchxpidm  8536  gchhar  8538  axgroth6  8695  hashsdom  11647  ruc  12834
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-br 4205  df-sdom 7104
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