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Theorem sdomentr 7170
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 7063 . 2  |-  ( B 
~~  C  ->  B  ~<_  C )
2 sdomdomtr 7169 . 2  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  C )
31, 2sylan2 461 1  |-  ( ( A  ~<  B  /\  B  ~~  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   class class class wbr 4146    ~~ cen 7035    ~<_ cdom 7036    ~< csdm 7037
This theorem is referenced by:  sdomen2  7181  unxpdom2  7246  sucxpdom  7247  findcard3  7279  fofinf1o  7316  sdomsdomcardi  7784  cardsdomel  7787  cardmin2  7811  alephnbtwn2  7879  pwsdompw  8010  infdif2  8016  fin23lem27  8134  axcclem  8263  numthcor  8300  sdomsdomcard  8361  pwcfsdom  8384  cfpwsdom  8385  inawinalem  8490  inatsk  8579  r1tskina  8583  tskuni  8584  rucALT  12749  iunmbl2  19311  dirith2  21082  erdszelem10  24658  pellex  26582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041
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