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Theorem endom 7336
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 7334 . 2  |-  ~~  C_  ~<_
21ssbri 4334 1  |-  ( A 
~~  B  ->  A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4292    ~~ cen 7307    ~<_ cdom 7308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-f1o 5425  df-en 7311  df-dom 7312
This theorem is referenced by:  bren2  7340  domrefg  7344  endomtr  7367  domentr  7368  domunsncan  7411  sbthb  7432  sdomentr  7445  ensdomtr  7447  domtriord  7457  domunsn  7461  xpen  7474  unxpdom2  7521  sucxpdom  7522  wdomen1  7791  wdomen2  7792  fidomtri2  8164  prdom2  8173  acnen  8223  acnen2  8225  alephdom  8251  alephinit  8265  uncdadom  8340  pwcdadom  8385  fin1a2lem11  8579  hsmexlem1  8595  gchdomtri  8796  gchcdaidm  8835  gchxpidm  8836  gchpwdom  8837  gchhar  8846  gruina  8985  odinf  16064  hauspwdom  19105  ufildom1  19499  iscmet3  20804  ovolctb2  20975  mbfaddlem  21138  nnct  26006  heiborlem3  28712
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