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Theorem sdomtr 6994
Description: Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
Assertion
Ref Expression
sdomtr  |-  ( ( A  ~<  B  /\  B  ~<  C )  ->  A  ~<  C )

Proof of Theorem sdomtr
StepHypRef Expression
1 sdomdom 6884 . 2  |-  ( A 
~<  B  ->  A  ~<_  B )
2 domsdomtr 6991 . 2  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<  C )
31, 2sylan 459 1  |-  ( ( A  ~<  B  /\  B  ~<  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   class class class wbr 4024    ~<_ cdom 6856    ~< csdm 6857
This theorem is referenced by:  sdomn2lp  6995  2pwuninel  7011  2pwne  7012  r1sdom  7441  alephordi  7696  pwsdompw  7825  gruina  8435  rexpen  12500
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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