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Theorem sdomentr 7011
Description: Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
sdomentr  |-  ( ( A  ~<  B  /\  B  ~~  C )  ->  A  ~<  C )

Proof of Theorem sdomentr
StepHypRef Expression
1 endom 6904 . 2  |-  ( B 
~~  C  ->  B  ~<_  C )
2 sdomdomtr 7010 . 2  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  C )
31, 2sylan2 460 1  |-  ( ( A  ~<  B  /\  B  ~~  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   class class class wbr 4039    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  sdomen2  7022  unxpdom2  7087  sucxpdom  7088  findcard3  7116  fofinf1o  7153  sdomsdomcardi  7620  cardsdomel  7623  cardmin2  7647  alephnbtwn2  7715  pwsdompw  7846  infdif2  7852  fin23lem27  7970  axcclem  8099  numthcor  8137  sdomsdomcard  8198  pwcfsdom  8221  cfpwsdom  8222  inawinalem  8327  inatsk  8416  r1tskina  8420  tskuni  8421  rucALT  12524  iunmbl2  18930  dirith2  20693  erdszelem10  23746  pellex  27023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
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