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Theorem bren2 7074
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 7070 . . 3  |-  ( A 
~~  B  ->  A  ~<_  B )
2 sdomnen 7072 . . . 4  |-  ( A 
~<  B  ->  -.  A  ~~  B )
32con2i 114 . . 3  |-  ( A 
~~  B  ->  -.  A  ~<  B )
41, 3jca 519 . 2  |-  ( A 
~~  B  ->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
5 brdom2 7073 . . . 4  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
65biimpi 187 . . 3  |-  ( A  ~<_  B  ->  ( A  ~<  B  \/  A  ~~  B ) )
76orcanai 880 . 2  |-  ( ( A  ~<_  B  /\  -.  A  ~<  B )  ->  A  ~~  B )
84, 7impbii 181 1  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359   class class class wbr 4153    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044
This theorem is referenced by:  marypha1lem  7373  tskwe  7770  infxpenlem  7828  cdainflem  8004  axcclem  8270  alephsuc3  8388  gchen1  8433  gchen2  8434  inatsk  8586  ufilen  17883  dirith2  21089
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 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-f1o 5401  df-en 7046  df-dom 7047  df-sdom 7048
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