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Theorem domsdomtr 6964
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 6857 . . 3  |-  ( B 
~<  C  ->  B  ~<_  C )
2 domtr 6882 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  C )  ->  A  ~<_  C )
31, 2sylan2 462 . 2  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<_  C )
4 simpr 449 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  B  ~<  C )
5 ensym 6878 . . . . . 6  |-  ( A 
~~  C  ->  C  ~~  A )
6 simpl 445 . . . . . 6  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<_  B )
7 endomtr 6887 . . . . . 6  |-  ( ( C  ~~  A  /\  A  ~<_  B )  ->  C  ~<_  B )
85, 6, 7syl2anr 466 . . . . 5  |-  ( ( ( A  ~<_  B  /\  B  ~<  C )  /\  A  ~~  C )  ->  C  ~<_  B )
9 domnsym 6955 . . . . 5  |-  ( C  ~<_  B  ->  -.  B  ~<  C )
108, 9syl 17 . . . 4  |-  ( ( ( A  ~<_  B  /\  B  ~<  C )  /\  A  ~~  C )  ->  -.  B  ~<  C )
1110ex 425 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  ( A  ~~  C  ->  -.  B  ~<  C ) )
124, 11mt2d 111 . 2  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  -.  A  ~~  C )
13 brsdom 6852 . 2  |-  ( A 
~<  C  <->  ( A  ~<_  C  /\  -.  A  ~~  C ) )
143, 12, 13sylanbrc 648 1  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   class class class wbr 3997    ~~ cen 6828    ~<_ cdom 6829    ~< csdm 6830
This theorem is referenced by:  ensdomtr  6965  sdomtr  6967  2pwuninel  6984  card2on  7236  tskwe  7551  harval2  7598  prdom2  7604  infxpenlem  7609  alephsucdom  7674  pwsdompw  7798  infunsdom1  7807  fin34  7984  ondomon  8153  cardmin  8154  konigthlem  8158  gchpwdom  8264  gchina  8289  inar1  8365  tskord  8370  tskuni  8373  tskurn  8379  csdfil  17551
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834
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