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Theorem endom 7125
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 7123 . 2  |-  ~~  C_  ~<_
21ssbri 4246 1  |-  ( A 
~~  B  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 4204    ~~ cen 7097    ~<_ cdom 7098
This theorem is referenced by:  bren2  7129  domrefg  7133  endomtr  7156  domentr  7157  domunsncan  7199  sbthb  7219  sdomentr  7232  ensdomtr  7234  domtriord  7244  domunsn  7248  xpen  7261  unxpdom2  7308  sucxpdom  7309  wdomen1  7533  wdomen2  7534  fidomtri2  7870  prdom2  7879  acnen  7923  acnen2  7925  alephdom  7951  alephinit  7965  uncdadom  8040  pwcdadom  8085  fin1a2lem11  8279  hsmexlem1  8295  gchdomtri  8493  gchcdaidm  8532  gchxpidm  8533  gchhar  8535  gchpwdom  8538  gruina  8682  odinf  15187  hauspwdom  17552  ufildom1  17946  iscmet3  19234  ovolctb2  19376  mbfaddlem  19540  nnct  24087  heiborlem3  26459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-f1o 5452  df-en 7101  df-dom 7102
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