MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domsdomtr Unicode version

Theorem domsdomtr 6992
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 6885 . . 3  |-  ( B 
~<  C  ->  B  ~<_  C )
2 domtr 6910 . . 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 6906 . . . . . 6  |-  ( A 
~~  C  ->  C  ~~  A )
6 simpl 445 . . . . . 6  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<_  B )
7 endomtr 6915 . . . . . 6  |-  ( ( C  ~~  A  /\  A  ~<_  B )  ->  C  ~<_  B )
85, 6, 7syl2anr 466 . . . . 5  |-  ( ( ( A  ~<_  B  /\  B  ~<  C )  /\  A  ~~  C )  ->  C  ~<_  B )
9 domnsym 6983 . . . . 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 6880 . 2  |-  ( A 
~<  C  <->  ( A  ~<_  C  /\  -.  A  ~~  C ) )
143, 12, 13sylanbrc 647 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 4025    ~~ cen 6856    ~<_ cdom 6857    ~< csdm 6858
This theorem is referenced by:  ensdomtr  6993  sdomtr  6995  2pwuninel  7012  card2on  7264  tskwe  7579  harval2  7626  prdom2  7632  infxpenlem  7637  alephsucdom  7702  pwsdompw  7826  infunsdom1  7835  fin34  8012  ondomon  8181  cardmin  8182  konigthlem  8186  gchpwdom  8292  gchina  8317  inar1  8393  tskord  8398  tskuni  8401  tskurn  8407  csdfil  17584
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  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 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862
  Copyright terms: Public domain W3C validator