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Theorem sdom0OLD 9060
Description: Obsolete version of sdom0 9059 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 8897 . . . 4 Rel ≺
21brrelex1i 5693 . . 3 (𝐴 ≺ ∅ → 𝐴 ∈ V)
3 0domg 9051 . . 3 (𝐴 ∈ V → ∅ ≼ 𝐴)
42, 3syl 17 . 2 (𝐴 ≺ ∅ → ∅ ≼ 𝐴)
5 domnsym 9050 . . 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 2107  Vcvv 3448  c0 4287   class class class wbr 5110  cdom 8888  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-11 2155  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-nfc 2890  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-er 8655  df-en 8891  df-dom 8892  df-sdom 8893
This theorem is referenced by: (None)
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