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Theorem endom 6904
Description: Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
Assertion
Ref Expression
endom  |-  ( A 
~~  B  ->  A  ~<_  B )

Proof of Theorem endom
StepHypRef Expression
1 enssdom 6902 . 2  |-  ~~  C_  ~<_
21ssbri 4081 1  |-  ( A 
~~  B  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 4039    ~~ cen 6876    ~<_ cdom 6877
This theorem is referenced by:  bren2  6908  domrefg  6912  endomtr  6935  domentr  6936  domunsncan  6978  sbthb  6998  sdomentr  7011  ensdomtr  7013  domtriord  7023  domunsn  7027  xpen  7040  unxpdom2  7087  sucxpdom  7088  wdomen1  7306  wdomen2  7307  fidomtri2  7643  prdom2  7652  acnen  7696  acnen2  7698  alephdom  7724  alephinit  7738  uncdadom  7813  pwcdadom  7858  fin1a2lem11  8052  hsmexlem1  8068  gchdomtri  8267  gchcdaidm  8306  gchxpidm  8307  gchhar  8309  gchpwdom  8312  gruina  8456  odinf  14892  hauspwdom  17243  ufildom1  17637  iscmet3  18735  ovolctb2  18867  mbfaddlem  19031  nnct  23350  heiborlem3  26640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-f1o 5278  df-en 6880  df-dom 6881
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