MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0we1 Unicode version

Theorem 0we1 6505
Description: The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)
Assertion
Ref Expression
0we1  |-  (/)  We  1o

Proof of Theorem 0we1
StepHypRef Expression
1 noel 3459 . . . 4  |-  -.  <. (/)
,  (/) >.  e.  (/)
2 df-br 4024 . . . 4  |-  ( (/) (/) (/) 
<-> 
<. (/) ,  (/) >.  e.  (/) )
31, 2mtbir 290 . . 3  |-  -.  (/) (/) (/)
4 rel0 4810 . . . 4  |-  Rel  (/)
5 wesn 4761 . . . 4  |-  ( Rel  (/)  ->  ( (/)  We  { (/)
}  <->  -.  (/) (/) (/) ) )
64, 5ax-mp 8 . . 3  |-  ( (/)  We 
{ (/) }  <->  -.  (/) (/) (/) )
73, 6mpbir 200 . 2  |-  (/)  We  { (/)
}
8 df1o2 6491 . . 3  |-  1o  =  { (/) }
9 weeq2 4382 . . 3  |-  ( 1o  =  { (/) }  ->  (
(/)  We  1o  <->  (/)  We  { (/)
} ) )
108, 9ax-mp 8 . 2  |-  ( (/)  We  1o  <->  (/)  We  { (/) } )
117, 10mpbir 200 1  |-  (/)  We  1o
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623    e. wcel 1684   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023    We wwe 4351   Rel wrel 4694   1oc1o 6472
This theorem is referenced by:  psr1tos  16268
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-suc 4398  df-xp 4695  df-rel 4696  df-1o 6479
  Copyright terms: Public domain W3C validator