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Theorem bren2 7130
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 7126 . . 3  |-  ( A 
~~  B  ->  A  ~<_  B )
2 sdomnen 7128 . . . 4  |-  ( A 
~<  B  ->  -.  A  ~~  B )
32con2i 114 . . 3  |-  ( A 
~~  B  ->  -.  A  ~<  B )
41, 3jca 519 . 2  |-  ( A 
~~  B  ->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
5 brdom2 7129 . . . 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 4204    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100
This theorem is referenced by:  marypha1lem  7430  tskwe  7829  infxpenlem  7887  cdainflem  8063  axcclem  8329  alephsuc3  8447  gchen1  8492  gchen2  8493  inatsk  8645  ufilen  17954  dirith2  21214  mblfinlem  26234
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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