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Theorem sdomirr 7237
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 7129 . . 3  |-  ( A 
~<  A  ->  -.  A  ~~  A )
2 enrefg 7132 . . 3  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2nsyl3 113 . 2  |-  ( A  e.  _V  ->  -.  A  ~<  A )
4 relsdom 7109 . . . 4  |-  Rel  ~<
54brrelexi 4911 . . 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 1725   _Vcvv 2949   class class class wbr 4205    ~~ cen 7099    ~< csdm 7101
This theorem is referenced by:  sdomn2lp  7239  2pwuninel  7255  2pwne  7256  r111  7694  alephval2  8440  alephom  8453  csdfil  17919
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 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-en 7103  df-dom 7104  df-sdom 7105
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