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Theorem sdomnen 6890
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 6884 . 2  |-  ( A 
~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
21simprbi 450 1  |-  ( A 
~<  B  ->  -.  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  bren2  6892  domdifsn  6945  sdomnsym  6986  domnsym  6987  sdomirr  6998  php5  7049  sucdom2  7057  pssinf  7073  f1finf1o  7086  isfinite2  7115  cardom  7619  pm54.43  7633  pr2ne  7635  alephdom  7708  cdainflem  7817  ackbij1b  7865  isfin4-3  7941  fin23lem25  7950  fin67  8021  axcclem  8083  canthp1lem2  8275  gchinf  8279  pwfseqlem4  8284  tskssel  8379  1nprm  12763  en2top  16723  carinttar  25902
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-br 4024  df-sdom 6866
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