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Theorem endom 8255
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 8253 . 2 ≈ ⊆ ≼
21ssbri 4920 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 4875  cen 8225  cdom 8226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-f1o 6134  df-en 8229  df-dom 8230
This theorem is referenced by:  bren2  8259  domrefg  8263  endomtr  8286  domentr  8287  domunsncan  8335  sbthb  8356  sdomentr  8369  ensdomtr  8371  domtriord  8381  domunsn  8385  xpen  8398  unxpdom2  8443  sucxpdom  8444  wdomen1  8757  wdomen2  8758  fidomtri2  9140  prdom2  9149  acnen  9196  acnen2  9198  alephdom  9224  alephinit  9238  uncdadom  9315  pwcdadom  9360  fin1a2lem11  9554  hsmexlem1  9570  gchdomtri  9773  gchcdaidm  9812  gchxpidm  9813  gchpwdom  9814  gchhar  9823  gruina  9962  nnct  13082  odinf  18338  hauspwdom  21682  ufildom1  22107  iscmet3  23468  ovolctb2  23665  mbfaddlem  23833  heiborlem3  34149  zct  40041  qct  40369  caratheodory  41530
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