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

Theorem sdomirr 7014
Description: Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
Assertion
Ref Expression
sdomirr  |-  -.  A  ~<  A

Proof of Theorem sdomirr
StepHypRef Expression
1 sdomnen 6906 . . 3  |-  ( A 
~<  A  ->  -.  A  ~~  A )
2 enrefg 6909 . . 3  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2nsyl3 111 . 2  |-  ( A  e.  _V  ->  -.  A  ~<  A )
4 relsdom 6886 . . . 4  |-  Rel  ~<
54brrelexi 4745 . . 3  |-  ( A 
~<  A  ->  A  e. 
_V )
65con3i 127 . 2  |-  ( -.  A  e.  _V  ->  -.  A  ~<  A )
73, 6pm2.61i 156 1  |-  -.  A  ~<  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    ~~ cen 6876    ~< csdm 6878
This theorem is referenced by:  sdomn2lp  7016  2pwuninel  7032  2pwne  7033  r111  7463  alephval2  8210  alephom  8223  csdfil  17605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-en 6880  df-dom 6881  df-sdom 6882
  Copyright terms: Public domain W3C validator