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Theorem endom 7929
Description: Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
Assertion
Ref Expression
endom (𝐴𝐵𝐴𝐵)

Proof of Theorem endom
StepHypRef Expression
1 enssdom 7927 . 2 ≈ ⊆ ≼
21ssbri 4659 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 4615  cen 7899  cdom 7900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-xp 5082  df-rel 5083  df-f1o 5856  df-en 7903  df-dom 7904
This theorem is referenced by:  bren2  7933  domrefg  7937  endomtr  7961  domentr  7962  domunsncan  8007  sbthb  8028  sdomentr  8041  ensdomtr  8043  domtriord  8053  domunsn  8057  xpen  8070  unxpdom2  8115  sucxpdom  8116  wdomen1  8428  wdomen2  8429  fidomtri2  8767  prdom2  8776  acnen  8823  acnen2  8825  alephdom  8851  alephinit  8865  uncdadom  8940  pwcdadom  8985  fin1a2lem11  9179  hsmexlem1  9195  gchdomtri  9398  gchcdaidm  9437  gchxpidm  9438  gchpwdom  9439  gchhar  9448  gruina  9587  nnct  12723  odinf  17904  hauspwdom  21217  ufildom1  21643  iscmet3  23004  ovolctb2  23173  mbfaddlem  23340  heiborlem3  33265  zct  38733  qct  39060  caratheodory  40065
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