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Theorem bren2 6892
Description: Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
bren2  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )

Proof of Theorem bren2
StepHypRef Expression
1 endom 6888 . . 3  |-  ( A 
~~  B  ->  A  ~<_  B )
2 sdomnen 6890 . . . 4  |-  ( A 
~<  B  ->  -.  A  ~~  B )
32con2i 112 . . 3  |-  ( A 
~~  B  ->  -.  A  ~<  B )
41, 3jca 518 . 2  |-  ( A 
~~  B  ->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
5 brdom2 6891 . . . 4  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
65biimpi 186 . . 3  |-  ( A  ~<_  B  ->  ( A  ~<  B  \/  A  ~~  B ) )
76orcanai 879 . 2  |-  ( ( A  ~<_  B  /\  -.  A  ~<  B )  ->  A  ~~  B )
84, 7impbii 180 1  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  marypha1lem  7186  tskwe  7583  infxpenlem  7641  cdainflem  7817  axcclem  8083  alephsuc3  8202  gchen1  8247  gchen2  8248  inatsk  8400  ufilen  17625  dirith2  20677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-f1o 5262  df-en 6864  df-dom 6865  df-sdom 6866
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