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Theorem sdomirr 7173
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 7065 . . 3  |-  ( A 
~<  A  ->  -.  A  ~~  A )
2 enrefg 7068 . . 3  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2nsyl3 113 . 2  |-  ( A  e.  _V  ->  -.  A  ~<  A )
4 relsdom 7045 . . . 4  |-  Rel  ~<
54brrelexi 4851 . . 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 1717   _Vcvv 2892   class class class wbr 4146    ~~ cen 7035    ~< csdm 7037
This theorem is referenced by:  sdomn2lp  7175  2pwuninel  7191  2pwne  7192  r111  7627  alephval2  8373  alephom  8386  csdfil  17840
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-en 7039  df-dom 7040  df-sdom 7041
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