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Theorem sdomirr 9065
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 8928 . . 3 (𝐴𝐴 → ¬ 𝐴𝐴)
2 enrefg 8931 . . 3 (𝐴 ∈ V → 𝐴𝐴)
31, 2nsyl3 138 . 2 (𝐴 ∈ V → ¬ 𝐴𝐴)
4 relsdom 8897 . . . 4 Rel ≺
54brrelex1i 5693 . . 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 2107  Vcvv 3448   class class class wbr 5110  cen 8887  csdm 8889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-en 8891  df-dom 8892  df-sdom 8893
This theorem is referenced by:  sdomn2lp  9067  2pwuninel  9083  2pwne  9084  r111  9718  alephval2  10515  alephom  10528  csdfil  23261
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