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Theorem sdomirr 7211
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 7103 . . 3  |-  ( A 
~<  A  ->  -.  A  ~~  A )
2 enrefg 7106 . . 3  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2nsyl3 113 . 2  |-  ( A  e.  _V  ->  -.  A  ~<  A )
4 relsdom 7083 . . . 4  |-  Rel  ~<
54brrelexi 4885 . . 3  |-  ( A 
~<  A  ->  A  e. 
_V )
65con3i 129 . 2  |-  ( -.  A  e.  _V  ->  -.  A  ~<  A )
73, 6pm2.61i 158 1  |-  -.  A  ~<  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1721   _Vcvv 2924   class class class wbr 4180    ~~ cen 7073    ~< csdm 7075
This theorem is referenced by:  sdomn2lp  7213  2pwuninel  7229  2pwne  7230  r111  7665  alephval2  8411  alephom  8424  csdfil  17887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-en 7077  df-dom 7078  df-sdom 7079
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