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Theorem sdomirr 7000
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 6892 . . 3  |-  ( A 
~<  A  ->  -.  A  ~~  A )
2 enrefg 6895 . . 3  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2nsyl3 111 . 2  |-  ( A  e.  _V  ->  -.  A  ~<  A )
4 relsdom 6872 . . . 4  |-  Rel  ~<
54brrelexi 4731 . . 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 1686   _Vcvv 2790   class class class wbr 4025    ~~ cen 6862    ~< csdm 6864
This theorem is referenced by:  sdomn2lp  7002  2pwuninel  7018  2pwne  7019  r111  7449  alephval2  8196  alephom  8209  csdfil  17591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  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 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-en 6866  df-dom 6867  df-sdom 6868
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