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Theorem endom 6888
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 6886 . 2  |-  ~~  C_  ~<_
21ssbri 4065 1  |-  ( A 
~~  B  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  bren2  6892  domrefg  6896  endomtr  6919  domentr  6920  domunsncan  6962  sbthb  6982  sdomentr  6995  ensdomtr  6997  domtriord  7007  domunsn  7011  xpen  7024  unxpdom2  7071  sucxpdom  7072  wdomen1  7290  wdomen2  7291  fidomtri2  7627  prdom2  7636  acnen  7680  acnen2  7682  alephdom  7708  alephinit  7722  uncdadom  7797  pwcdadom  7842  fin1a2lem11  8036  hsmexlem1  8052  gchdomtri  8251  gchcdaidm  8290  gchxpidm  8291  gchhar  8293  gchpwdom  8296  gruina  8440  odinf  14876  hauspwdom  17227  ufildom1  17621  iscmet3  18719  ovolctb2  18851  mbfaddlem  19015  nnct  23335  heiborlem3  26537
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-f1o 5262  df-en 6864  df-dom 6865
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