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Theorem rngosn6 22018
Description: 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  e.  RingOps  ->  ( X  ~~  1o 
<->  R  =  <. { <. <. Z ,  Z >. ,  Z >. } ,  { <. <. Z ,  Z >. ,  Z >. } >. ) )

Proof of Theorem rngosn6
StepHypRef Expression
1 on1el3.1 . . 3  |-  G  =  ( 1st `  R
)
2 on1el3.2 . . 3  |-  X  =  ran  G
3 on1el3.3 . . 3  |-  Z  =  (GId `  G )
41, 2, 3rngo0cl 21988 . 2  |-  ( R  e.  RingOps  ->  Z  e.  X
)
51, 2rngosn4 22017 . 2  |-  ( ( R  e.  RingOps  /\  Z  e.  X )  ->  ( X  ~~  1o  <->  R  =  <. { <. <. Z ,  Z >. ,  Z >. } ,  { <. <. Z ,  Z >. ,  Z >. } >. ) )
64, 5mpdan 651 1  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  R  =  <. { <. <. Z ,  Z >. ,  Z >. } ,  { <. <. Z ,  Z >. ,  Z >. } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   {csn 3816   <.cop 3819   class class class wbr 4214   ran crn 4881   ` cfv 5456   1stc1st 6349   1oc1o 6719    ~~ cen 7108  GIdcgi 21777   RingOpscrngo 21965
This theorem is referenced by:  dvrunz  22023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-riota 6551  df-1o 6726  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-grpo 21781  df-gid 21782  df-ablo 21872  df-rngo 21966
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