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Theorem sdomnen 7095
Description: Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)
Assertion
Ref Expression
sdomnen  |-  ( A 
~<  B  ->  -.  A  ~~  B )

Proof of Theorem sdomnen
StepHypRef Expression
1 brsdom 7089 . 2  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
21simprbi 451 1  |-  ( A 
~<  B  ->  -.  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   class class class wbr 4172    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067
This theorem is referenced by:  bren2  7097  domdifsn  7150  sdomnsym  7191  domnsym  7192  sdomirr  7203  php5  7254  sucdom2  7262  pssinf  7278  f1finf1o  7294  isfinite2  7324  cardom  7829  pm54.43  7843  pr2ne  7845  alephdom  7918  cdainflem  8027  ackbij1b  8075  isfin4-3  8151  fin23lem25  8160  fin67  8231  axcclem  8293  canthp1lem2  8484  gchinf  8488  pwfseqlem4  8493  tskssel  8588  1nprm  13039  en2top  17005
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-dif 3283  df-br 4173  df-sdom 7071
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