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Theorem rngosn6 21111
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 21081 . 2  |-  ( R  e.  RingOps  ->  Z  e.  X
)
51, 2rngosn4 21110 . 2  |-  ( ( R  e.  RingOps  /\  Z  e.  X )  ->  ( X  ~~  1o  <->  R  =  <. { <. <. Z ,  Z >. ,  Z >. } ,  { <. <. Z ,  Z >. ,  Z >. } >. ) )
64, 5mpdan 649 1  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  R  =  <. { <. <. Z ,  Z >. ,  Z >. } ,  { <. <. Z ,  Z >. ,  Z >. } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656   class class class wbr 4039   ran crn 4706   ` cfv 5271   1stc1st 6136   1oc1o 6488    ~~ cen 6876  GIdcgi 20870   RingOpscrngo 21058
This theorem is referenced by:  dvrunz  21116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-grpo 20874  df-gid 20875  df-ablo 20965  df-rngo 21059
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