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Theorem domsdomtr 6881
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 6775 . . 3  |-  ( B 
~<  C  ->  B  ~<_  C )
2 domtr 6799 . . 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 6796 . . . . . 6  |-  ( A 
~~  C  ->  C  ~~  A )
6 simpl 445 . . . . . 6  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<_  B )
7 endomtr 6804 . . . . . 6  |-  ( ( C  ~~  A  /\  A  ~<_  B )  ->  C  ~<_  B )
85, 6, 7syl2anr 466 . . . . 5  |-  ( ( ( A  ~<_  B  /\  B  ~<  C )  /\  A  ~~  C )  ->  C  ~<_  B )
9 domnsym 6872 . . . . 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 6770 . 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 3920    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748
This theorem is referenced by:  ensdomtr  6882  sdomtr  6884  2pwuninel  6901  card2on  7152  tskwe  7467  harval2  7514  prdom2  7520  infxpenlem  7525  alephsucdom  7590  pwsdompw  7714  infunsdom1  7723  fin34  7900  ondomon  8067  cardmin  8068  konigthlem  8070  gchpwdom  8176  gchina  8201  inar1  8277  tskord  8282  tskuni  8285  tskurn  8291  csdfil  17421
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752
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