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Theorem sdom0OLD 9149
Description: Obsolete version of sdom0 9148 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sdom0OLD ¬ 𝐴 ≺ ∅

Proof of Theorem sdom0OLD
StepHypRef Expression
1 relsdom 8992 . . . 4 Rel ≺
21brrelex1i 5741 . . 3 (𝐴 ≺ ∅ → 𝐴 ∈ V)
3 0domg 9140 . . 3 (𝐴 ∈ V → ∅ ≼ 𝐴)
42, 3syl 17 . 2 (𝐴 ≺ ∅ → ∅ ≼ 𝐴)
5 domnsym 9139 . . 3 (∅ ≼ 𝐴 → ¬ 𝐴 ≺ ∅)
65con2i 139 . 2 (𝐴 ≺ ∅ → ¬ ∅ ≼ 𝐴)
74, 6pm2.65i 194 1 ¬ 𝐴 ≺ ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2108  Vcvv 3480  c0 4333   class class class wbr 5143  cdom 8983  csdm 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by: (None)
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