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Theorem endom 7544
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 7542 . 2  |-  ~~  C_  ~<_
21ssbri 4479 1  |-  ( A 
~~  B  ->  A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4437    ~~ cen 7515    ~<_ cdom 7516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-f1o 5585  df-en 7519  df-dom 7520
This theorem is referenced by:  bren2  7548  domrefg  7552  endomtr  7575  domentr  7576  domunsncan  7619  sbthb  7640  sdomentr  7653  ensdomtr  7655  domtriord  7665  domunsn  7669  xpen  7682  unxpdom2  7730  sucxpdom  7731  wdomen1  8005  wdomen2  8006  fidomtri2  8378  prdom2  8387  acnen  8437  acnen2  8439  alephdom  8465  alephinit  8479  uncdadom  8554  pwcdadom  8599  fin1a2lem11  8793  hsmexlem1  8809  gchdomtri  9010  gchcdaidm  9049  gchxpidm  9050  gchpwdom  9051  gchhar  9060  gruina  9199  odinf  16564  hauspwdom  19980  ufildom1  20405  iscmet3  21710  ovolctb2  21881  mbfaddlem  22045  nnct  27507  heiborlem3  30285
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