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Theorem domsdomtr 6996
Description: Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
domsdomtr  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<  C )

Proof of Theorem domsdomtr
StepHypRef Expression
1 sdomdom 6889 . . 3  |-  ( B 
~<  C  ->  B  ~<_  C )
2 domtr 6914 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  C )  ->  A  ~<_  C )
31, 2sylan2 460 . 2  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<_  C )
4 simpr 447 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  B  ~<  C )
5 ensym 6910 . . . . . 6  |-  ( A 
~~  C  ->  C  ~~  A )
6 simpl 443 . . . . . 6  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<_  B )
7 endomtr 6919 . . . . . 6  |-  ( ( C  ~~  A  /\  A  ~<_  B )  ->  C  ~<_  B )
85, 6, 7syl2anr 464 . . . . 5  |-  ( ( ( A  ~<_  B  /\  B  ~<  C )  /\  A  ~~  C )  ->  C  ~<_  B )
9 domnsym 6987 . . . . 5  |-  ( C  ~<_  B  ->  -.  B  ~<  C )
108, 9syl 15 . . . 4  |-  ( ( ( A  ~<_  B  /\  B  ~<  C )  /\  A  ~~  C )  ->  -.  B  ~<  C )
1110ex 423 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  ( A  ~~  C  ->  -.  B  ~<  C ) )
124, 11mt2d 109 . 2  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  -.  A  ~~  C )
13 brsdom 6884 . 2  |-  ( A 
~<  C  <->  ( A  ~<_  C  /\  -.  A  ~~  C ) )
143, 12, 13sylanbrc 645 1  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  ensdomtr  6997  sdomtr  6999  2pwuninel  7016  card2on  7268  tskwe  7583  harval2  7630  prdom2  7636  infxpenlem  7641  alephsucdom  7706  pwsdompw  7830  infunsdom1  7839  fin34  8016  ondomon  8185  cardmin  8186  konigthlem  8190  gchpwdom  8296  gchina  8321  inar1  8397  tskord  8402  tskuni  8405  tskurn  8411  csdfil  17589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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