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

Theorem 0we1 6521
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 3472 . . . 4  |-  -.  <. (/)
,  (/) >.  e.  (/)
2 df-br 4040 . . . 4  |-  ( (/) (/) (/) 
<-> 
<. (/) ,  (/) >.  e.  (/) )
31, 2mtbir 290 . . 3  |-  -.  (/) (/) (/)
4 rel0 4826 . . . 4  |-  Rel  (/)
5 wesn 4777 . . . 4  |-  ( Rel  (/)  ->  ( (/)  We  { (/)
}  <->  -.  (/) (/) (/) ) )
64, 5ax-mp 8 . . 3  |-  ( (/)  We 
{ (/) }  <->  -.  (/) (/) (/) )
73, 6mpbir 200 . 2  |-  (/)  We  { (/)
}
8 df1o2 6507 . . 3  |-  1o  =  { (/) }
9 weeq2 4398 . . 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 1632    e. wcel 1696   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039    We wwe 4367   Rel wrel 4710   1oc1o 6488
This theorem is referenced by:  psr1tos  16284
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-suc 4414  df-xp 4711  df-rel 4712  df-1o 6495
  Copyright terms: Public domain W3C validator