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Theorem endom 7461
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 7459 . 2  |-  ~~  C_  ~<_
21ssbri 4409 1  |-  ( A 
~~  B  ->  A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4367    ~~ cen 7432    ~<_ cdom 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-f1o 5503  df-en 7436  df-dom 7437
This theorem is referenced by:  bren2  7465  domrefg  7469  endomtr  7492  domentr  7493  domunsncan  7536  sbthb  7557  sdomentr  7570  ensdomtr  7572  domtriord  7582  domunsn  7586  xpen  7599  unxpdom2  7644  sucxpdom  7645  wdomen1  7917  wdomen2  7918  fidomtri2  8288  prdom2  8297  acnen  8347  acnen2  8349  alephdom  8375  alephinit  8389  uncdadom  8464  pwcdadom  8509  fin1a2lem11  8703  hsmexlem1  8719  gchdomtri  8918  gchcdaidm  8957  gchxpidm  8958  gchpwdom  8959  gchhar  8968  gruina  9107  odinf  16702  hauspwdom  20087  ufildom1  20512  iscmet3  21817  ovolctb2  21988  mbfaddlem  22152  nnct  27678  heiborlem3  30475
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