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Theorem sdom0OLD 9111
Description: Obsolete version of sdom0 9110 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 8948 . . . 4 Rel ≺
21brrelex1i 5725 . . 3 (𝐴 ≺ ∅ → 𝐴 ∈ V)
3 0domg 9102 . . 3 (𝐴 ∈ V → ∅ ≼ 𝐴)
42, 3syl 17 . 2 (𝐴 ≺ ∅ → ∅ ≼ 𝐴)
5 domnsym 9101 . . 3 (∅ ≼ 𝐴 → ¬ 𝐴 ≺ ∅)
65con2i 139 . 2 (𝐴 ≺ ∅ → ¬ ∅ ≼ 𝐴)
74, 6pm2.65i 193 1 ¬ 𝐴 ≺ ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2098  Vcvv 3468  c0 4317   class class class wbr 5141  cdom 8939  csdm 8940
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-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
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-nfc 2879  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 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944
This theorem is referenced by: (None)
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