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Theorem sdomdomtr 6927
Description: Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
sdomdomtr  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  C )

Proof of Theorem sdomdomtr
StepHypRef Expression
1 sdomdom 6822 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 domtr 6847 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  C )  ->  A  ~<_  C )
31, 2sylan 459 . 2  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<_  C )
4 simpl 445 . . 3  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  B )
5 simpr 449 . . . . . 6  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  B  ~<_  C )
6 ensym 6843 . . . . . 6  |-  ( A 
~~  C  ->  C  ~~  A )
7 domentr 6853 . . . . . 6  |-  ( ( B  ~<_  C  /\  C  ~~  A )  ->  B  ~<_  A )
85, 6, 7syl2an 465 . . . . 5  |-  ( ( ( A  ~<  B  /\  B  ~<_  C )  /\  A  ~~  C )  ->  B  ~<_  A )
9 domnsym 6920 . . . . 5  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
108, 9syl 17 . . . 4  |-  ( ( ( A  ~<  B  /\  B  ~<_  C )  /\  A  ~~  C )  ->  -.  A  ~<  B )
1110ex 425 . . 3  |-  ( ( A  ~<  B  /\  B  ~<_  C )  -> 
( A  ~~  C  ->  -.  A  ~<  B ) )
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:  sdomentr  6928  sucdom  6991  sucdomiOLD  6992  infsdomnn  7051  fodomfib  7069  marypha1lem  7119  r1sdom  7379  infxpenlem  7574  infunsdom1  7772  fin56  7952  fodomb  8084  pwcfsdom  8138  cfpwsdom  8139  canthp1lem2  8208  gchhar  8226  gchpwdom  8229  gchina  8254  tsksdom  8311  tskpr  8325  tskcard  8336  gruina  8373
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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