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Theorem sdomirr 9113
Description: Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
Assertion
Ref Expression
sdomirr ¬ 𝐴𝐴

Proof of Theorem sdomirr
StepHypRef Expression
1 sdomnen 8976 . . 3 (𝐴𝐴 → ¬ 𝐴𝐴)
2 enrefg 8979 . . 3 (𝐴 ∈ V → 𝐴𝐴)
31, 2nsyl3 138 . 2 (𝐴 ∈ V → ¬ 𝐴𝐴)
4 relsdom 8945 . . . 4 Rel ≺
54brrelex1i 5725 . . 3 (𝐴𝐴𝐴 ∈ V)
65con3i 154 . 2 𝐴 ∈ V → ¬ 𝐴𝐴)
73, 6pm2.61i 182 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2098  Vcvv 3468   class class class wbr 5141  cen 8935  csdm 8937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-en 8939  df-dom 8940  df-sdom 8941
This theorem is referenced by:  sdomn2lp  9115  2pwuninel  9131  2pwne  9132  r111  9769  alephval2  10566  alephom  10579  csdfil  23748
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