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Theorem sdomentr 6990
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 6883 . 2  |-  ( B 
~~  C  ->  B  ~<_  C )
2 sdomdomtr 6989 . 2  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  C )
31, 2sylan2 462 1  |-  ( ( A  ~<  B  /\  B  ~~  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   class class class wbr 4024    ~~ cen 6855    ~<_ cdom 6856    ~< csdm 6857
This theorem is referenced by:  sdomen2  7001  unxpdom2  7066  sucxpdom  7067  findcard3  7095  fofinf1o  7132  sdomsdomcardi  7599  cardsdomel  7602  cardmin2  7626  alephnbtwn2  7694  pwsdompw  7825  infdif2  7831  fin23lem27  7949  axcclem  8078  numthcor  8116  sdomsdomcard  8177  pwcfsdom  8200  cfpwsdom  8201  inawinalem  8306  inatsk  8395  r1tskina  8399  tskuni  8400  rucALT  12502  iunmbl2  18908  dirith2  20671  erdszelem10  23135  pellex  26319
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-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861
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