MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sdom0OLD Structured version   Visualization version   GIF version

Theorem sdom0OLD 9148
Description: Obsolete version of sdom0 9147 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 8991 . . . 4 Rel ≺
21brrelex1i 5745 . . 3 (𝐴 ≺ ∅ → 𝐴 ∈ V)
3 0domg 9139 . . 3 (𝐴 ∈ V → ∅ ≼ 𝐴)
42, 3syl 17 . 2 (𝐴 ≺ ∅ → ∅ ≼ 𝐴)
5 domnsym 9138 . . 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 2106  Vcvv 3478  c0 4339   class class class wbr 5148  cdom 8982  csdm 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator