Theorem rngosn6 33855

Description: Obsolete as of 25-Jan-2020. Use ringen1zr 19325 or srgen1zr 18576 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)

Hypotheses

Ref	Expression
on1el3.1	⊢ 𝐺 = (1st ‘𝑅)
on1el3.2	⊢ 𝑋 = ran 𝐺
on1el3.3	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
rngosn6	⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜 ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))

Proof of Theorem rngosn6

Step	Hyp	Ref	Expression
1		on1el3.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		on1el3.2	. . 3 ⊢ 𝑋 = ran 𝐺
3		on1el3.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
4	1, 2, 3	rngo0cl 33848	. 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
5	1, 2	rngosn4 33854	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋) → (𝑋 ≈ 1𝑜 ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
6	4, 5	mpdan 703	1 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜 ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523   ∈ wcel 2030  {csn 4210  ⟨cop 4216   class class class wbr 4685  ran crn 5144  ‘cfv 5926  1^st c1st 7208  1_𝑜c1o 7598   ≈ cen 7994  GIdcgi 27472  RingOpscrngo 33823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-om 7108  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-grpo 27475  df-gid 27476  df-ablo 27527  df-rngo 33824

This theorem is referenced by:  dvrunz  33883