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Theorem endom 7539
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 7537 . 2  |-  ~~  C_  ~<_
21ssbri 4489 1  |-  ( A 
~~  B  ->  A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4447    ~~ cen 7510    ~<_ cdom 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-f1o 5593  df-en 7514  df-dom 7515
This theorem is referenced by:  bren2  7543  domrefg  7547  endomtr  7570  domentr  7571  domunsncan  7614  sbthb  7635  sdomentr  7648  ensdomtr  7650  domtriord  7660  domunsn  7664  xpen  7677  unxpdom2  7725  sucxpdom  7726  wdomen1  7998  wdomen2  7999  fidomtri2  8371  prdom2  8380  acnen  8430  acnen2  8432  alephdom  8458  alephinit  8472  uncdadom  8547  pwcdadom  8592  fin1a2lem11  8786  hsmexlem1  8802  gchdomtri  9003  gchcdaidm  9042  gchxpidm  9043  gchpwdom  9044  gchhar  9053  gruina  9192  odinf  16378  hauspwdom  19765  ufildom1  20159  iscmet3  21464  ovolctb2  21635  mbfaddlem  21799  nnct  27198  heiborlem3  29910
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