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Theorem sdom1OLD 9279
Description: Obsolete version of sdom1 9278 as of 12-Dec-2024. (Contributed by Stefan O'Rear, 28-Oct-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sdom1OLD (𝐴 ≺ 1o𝐴 = ∅)

Proof of Theorem sdom1OLD
StepHypRef Expression
1 domnsym 9139 . . . . 5 (1o𝐴 → ¬ 𝐴 ≺ 1o)
21con2i 139 . . . 4 (𝐴 ≺ 1o → ¬ 1o𝐴)
3 0sdom1dom 9274 . . . 4 (∅ ≺ 𝐴 ↔ 1o𝐴)
42, 3sylnibr 329 . . 3 (𝐴 ≺ 1o → ¬ ∅ ≺ 𝐴)
5 relsdom 8992 . . . . 5 Rel ≺
65brrelex1i 5741 . . . 4 (𝐴 ≺ 1o𝐴 ∈ V)
7 0sdomg 9144 . . . . 5 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
87necon2bbid 2984 . . . 4 (𝐴 ∈ V → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴))
96, 8syl 17 . . 3 (𝐴 ≺ 1o → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴))
104, 9mpbird 257 . 2 (𝐴 ≺ 1o𝐴 = ∅)
11 1n0 8526 . . . 4 1o ≠ ∅
12 1oex 8516 . . . . 5 1o ∈ V
13120sdom 9147 . . . 4 (∅ ≺ 1o ↔ 1o ≠ ∅)
1411, 13mpbir 231 . . 3 ∅ ≺ 1o
15 breq1 5146 . . 3 (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o))
1614, 15mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 ≺ 1o)
1710, 16impbii 209 1 (𝐴 ≺ 1o𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  c0 4333   class class class wbr 5143  1oc1o 8499  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-ne 2941  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-suc 6390  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by: (None)
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