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Theorem rngosn4 21094
Description: 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  =  ( 1st `  R
)
on1el3.2  |-  X  =  ran  G
Assertion
Ref Expression
rngosn4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( X  ~~  1o  <->  R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >. ) )

Proof of Theorem rngosn4
StepHypRef Expression
1 en1eqsn 7088 . . . . 5  |-  ( ( A  e.  X  /\  X  ~~  1o )  ->  X  =  { A } )
21ex 423 . . . 4  |-  ( A  e.  X  ->  ( X  ~~  1o  ->  X  =  { A } ) )
3 ensn1g 6926 . . . . 5  |-  ( A  e.  X  ->  { A }  ~~  1o )
4 breq1 4026 . . . . 5  |-  ( X  =  { A }  ->  ( X  ~~  1o  <->  { A }  ~~  1o ) )
53, 4syl5ibrcom 213 . . . 4  |-  ( A  e.  X  ->  ( X  =  { A }  ->  X  ~~  1o ) )
62, 5impbid 183 . . 3  |-  ( A  e.  X  ->  ( X  ~~  1o  <->  X  =  { A } ) )
76adantl 452 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( X  ~~  1o  <->  X  =  { A } ) )
8 on1el3.1 . . 3  |-  G  =  ( 1st `  R
)
9 on1el3.2 . . 3  |-  X  =  ran  G
108, 9rngosn3 21093 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( X  =  { A } 
<->  R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >. ) )
117, 10bitrd 244 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( X  ~~  1o  <->  R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643   class class class wbr 4023   ran crn 4690   ` cfv 5255   1stc1st 6120   1oc1o 6472    ~~ cen 6860   RingOpscrngo 21042
This theorem is referenced by:  rngosn6  21095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-grpo 20858  df-ablo 20949  df-rngo 21043
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