MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngosn4 Structured version   Unicode version

Theorem rngosn4 22015
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 7338 . . . . 5  |-  ( ( A  e.  X  /\  X  ~~  1o )  ->  X  =  { A } )
21ex 424 . . . 4  |-  ( A  e.  X  ->  ( X  ~~  1o  ->  X  =  { A } ) )
3 ensn1g 7172 . . . . 5  |-  ( A  e.  X  ->  { A }  ~~  1o )
4 breq1 4215 . . . . 5  |-  ( X  =  { A }  ->  ( X  ~~  1o  <->  { A }  ~~  1o ) )
53, 4syl5ibrcom 214 . . . 4  |-  ( A  e.  X  ->  ( X  =  { A }  ->  X  ~~  1o ) )
62, 5impbid 184 . . 3  |-  ( A  e.  X  ->  ( X  ~~  1o  <->  X  =  { A } ) )
76adantl 453 . 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 22014 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( X  =  { A } 
<->  R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >. ) )
117, 10bitrd 245 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814   <.cop 3817   class class class wbr 4212   ran crn 4879   ` cfv 5454   1stc1st 6347   1oc1o 6717    ~~ cen 7106   RingOpscrngo 21963
This theorem is referenced by:  rngosn6  22016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-grpo 21779  df-ablo 21870  df-rngo 21964
  Copyright terms: Public domain W3C validator