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Theorem bren2 7618
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 7614 . . 3  |-  ( A 
~~  B  ->  A  ~<_  B )
2 sdomnen 7616 . . . 4  |-  ( A 
~<  B  ->  -.  A  ~~  B )
32con2i 124 . . 3  |-  ( A 
~~  B  ->  -.  A  ~<  B )
41, 3jca 541 . 2  |-  ( A 
~~  B  ->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
5 brdom2 7617 . . . 4  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
65biimpi 199 . . 3  |-  ( A  ~<_  B  ->  ( A  ~<  B  \/  A  ~~  B ) )
76orcanai 927 . 2  |-  ( ( A  ~<_  B  /\  -.  A  ~<  B )  ->  A  ~~  B )
84, 7impbii 192 1  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    \/ wo 375    /\ wa 376   class class class wbr 4395    ~~ cen 7584    ~<_ cdom 7585    ~< csdm 7586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-f1o 5596  df-en 7588  df-dom 7589  df-sdom 7590
This theorem is referenced by:  marypha1lem  7965  tskwe  8402  infxpenlem  8462  cdainflem  8639  axcclem  8905  alephsuc3  9023  gchen1  9068  gchen2  9069  inatsk  9221  ufilen  21023  dirith2  24445  f1ocnt  28451  mblfinlem1  32041
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