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Theorem sdomirr 6994
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 6886 . . 3  |-  ( A 
~<  A  ->  -.  A  ~~  A )
2 enrefg 6889 . . 3  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2nsyl3 111 . 2  |-  ( A  e.  _V  ->  -.  A  ~<  A )
4 relsdom 6866 . . . 4  |-  Rel  ~<
54brrelexi 4728 . . 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 1685   _Vcvv 2789   class class class wbr 4024    ~~ cen 6856    ~< csdm 6858
This theorem is referenced by:  sdomn2lp  6996  2pwuninel  7012  2pwne  7013  r111  7443  alephval2  8190  alephom  8203  csdfil  17585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-en 6860  df-dom 6861  df-sdom 6862
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