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

Theorem sdomtr 7236
Description: Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
Assertion
Ref Expression
sdomtr  |-  ( ( A  ~<  B  /\  B  ~<  C )  ->  A  ~<  C )

Proof of Theorem sdomtr
StepHypRef Expression
1 sdomdom 7126 . 2  |-  ( A 
~<  B  ->  A  ~<_  B )
2 domsdomtr 7233 . 2  |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<  C )
31, 2sylan 458 1  |-  ( ( A  ~<  B  /\  B  ~<  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   class class class wbr 4204    ~<_ cdom 7098    ~< csdm 7099
This theorem is referenced by:  sdomn2lp  7237  2pwuninel  7253  2pwne  7254  r1sdom  7689  alephordi  7944  pwsdompw  8073  gruina  8682  rexpen  12815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103
  Copyright terms: Public domain W3C validator