MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  endom Unicode version

Theorem endom 6884
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 6882 . 2  |-  ~~  C_  ~<_
21ssbri 4067 1  |-  ( A 
~~  B  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6   class class class wbr 4025    ~~ cen 6856    ~<_ cdom 6857
This theorem is referenced by:  bren2  6888  domrefg  6892  endomtr  6915  domentr  6916  domunsncan  6958  sbthb  6978  sdomentr  6991  ensdomtr  6993  domtriord  7003  domunsn  7007  xpen  7020  unxpdom2  7067  sucxpdom  7068  wdomen1  7286  wdomen2  7287  fidomtri2  7623  prdom2  7632  acnen  7676  acnen2  7678  alephdom  7704  alephinit  7718  uncdadom  7793  pwcdadom  7838  fin1a2lem11  8032  hsmexlem1  8048  gchdomtri  8247  gchcdaidm  8286  gchxpidm  8287  gchhar  8289  gchpwdom  8292  gruina  8436  odinf  14871  hauspwdom  17222  ufildom1  17616  iscmet3  18714  ovolctb2  18846  mbfaddlem  19010  heiborlem3  25937
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4695  df-rel 4696  df-f1o 5229  df-en 6860  df-dom 6861
  Copyright terms: Public domain W3C validator