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Theorem rngosn4 35846
Description: Obsolete as of 25-Jan-2020. Use rngen1zr 20338 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
on1el3.1 𝐺 = (1st𝑅)
on1el3.2 𝑋 = ran 𝐺
Assertion
Ref Expression
rngosn4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Proof of Theorem rngosn4
StepHypRef Expression
1 en1eqsnbi 8929 . . 3 (𝐴𝑋 → (𝑋 ≈ 1o𝑋 = {𝐴}))
21adantl 485 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 ≈ 1o𝑋 = {𝐴}))
3 on1el3.1 . . 3 𝐺 = (1st𝑅)
4 on1el3.2 . . 3 𝑋 = ran 𝐺
53, 4rngosn3 35845 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
62, 5bitrd 282 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  {csn 4555  cop 4561   class class class wbr 5067  ran crn 5566  cfv 6397  1st c1st 7777  1oc1o 8215  cen 8643  RingOpscrngo 35815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pow 5272  ax-pr 5336  ax-un 7541
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3422  df-sbc 3709  df-csb 3826  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-pss 3899  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4834  df-iun 4920  df-br 5068  df-opab 5130  df-mpt 5150  df-tr 5176  df-id 5469  df-eprel 5474  df-po 5482  df-so 5483  df-fr 5523  df-we 5525  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-ord 6233  df-on 6234  df-lim 6235  df-suc 6236  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404  df-fv 6405  df-ov 7234  df-om 7663  df-1st 7779  df-2nd 7780  df-1o 8222  df-er 8411  df-en 8647  df-dom 8648  df-sdom 8649  df-fin 8650  df-grpo 28598  df-ablo 28650  df-rngo 35816
This theorem is referenced by:  rngosn6  35847
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