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Theorem sdom1OLD 9277
Description: Obsolete version of sdom1 9276 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 9138 . . . . 5 (1o𝐴 → ¬ 𝐴 ≺ 1o)
21con2i 139 . . . 4 (𝐴 ≺ 1o → ¬ 1o𝐴)
3 0sdom1dom 9272 . . . 4 (∅ ≺ 𝐴 ↔ 1o𝐴)
42, 3sylnibr 329 . . 3 (𝐴 ≺ 1o → ¬ ∅ ≺ 𝐴)
5 relsdom 8991 . . . . 5 Rel ≺
65brrelex1i 5745 . . . 4 (𝐴 ≺ 1o𝐴 ∈ V)
7 0sdomg 9143 . . . . 5 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
87necon2bbid 2982 . . . 4 (𝐴 ∈ V → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴))
96, 8syl 17 . . 3 (𝐴 ≺ 1o → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴))
104, 9mpbird 257 . 2 (𝐴 ≺ 1o𝐴 = ∅)
11 1n0 8525 . . . 4 1o ≠ ∅
12 1oex 8515 . . . . 5 1o ∈ V
13120sdom 9146 . . . 4 (∅ ≺ 1o ↔ 1o ≠ ∅)
1411, 13mpbir 231 . . 3 ∅ ≺ 1o
15 breq1 5151 . . 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 1537  wcel 2106  wne 2938  Vcvv 3478  c0 4339   class class class wbr 5148  1oc1o 8498  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-ne 2939  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-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987
This theorem is referenced by: (None)
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