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Theorem sdomirr 8094
 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 7981 . . 3 (𝐴𝐴 → ¬ 𝐴𝐴)
2 enrefg 7984 . . 3 (𝐴 ∈ V → 𝐴𝐴)
31, 2nsyl3 133 . 2 (𝐴 ∈ V → ¬ 𝐴𝐴)
4 relsdom 7959 . . . 4 Rel ≺
54brrelexi 5156 . . 3 (𝐴𝐴𝐴 ∈ V)
65con3i 150 . 2 𝐴 ∈ V → ¬ 𝐴𝐴)
73, 6pm2.61i 176 1 ¬ 𝐴𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1989  Vcvv 3198   class class class wbr 4651   ≈ cen 7949   ≺ csdm 7951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-en 7953  df-dom 7954  df-sdom 7955 This theorem is referenced by:  sdomn2lp  8096  2pwuninel  8112  2pwne  8113  r111  8635  alephval2  9391  alephom  9404  csdfil  21692
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