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Theorem domsdomtr 6929
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 6822 . . 3  |-  ( B 
~<  C  ->  B  ~<_  C )
2 domtr 6847 . . 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 6843 . . . . . 6  |-  ( A 
~~  C  ->  C  ~~  A )
6 simpl 445 . . . . . 6  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<_  B )
7 endomtr 6852 . . . . . 6  |-  ( ( C  ~~  A  /\  A  ~<_  B )  ->  C  ~<_  B )
85, 6, 7syl2anr 466 . . . . 5  |-  ( ( ( A  ~<_  B  /\  B  ~<  C )  /\  A  ~~  C )  ->  C  ~<_  B )
9 domnsym 6920 . . . . 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 6817 . 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 3963    ~~ cen 6793    ~<_ cdom 6794    ~< csdm 6795
This theorem is referenced by:  ensdomtr  6930  sdomtr  6932  2pwuninel  6949  card2on  7201  tskwe  7516  harval2  7563  prdom2  7569  infxpenlem  7574  alephsucdom  7639  pwsdompw  7763  infunsdom1  7772  fin34  7949  ondomon  8118  cardmin  8119  konigthlem  8123  gchpwdom  8229  gchina  8254  inar1  8330  tskord  8335  tskuni  8338  tskurn  8344  csdfil  17516
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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799
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