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Theorem 0we1 6687
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 3576 . . . 4  |-  -.  <. (/)
,  (/) >.  e.  (/)
2 df-br 4155 . . . 4  |-  ( (/) (/) (/) 
<-> 
<. (/) ,  (/) >.  e.  (/) )
31, 2mtbir 291 . . 3  |-  -.  (/) (/) (/)
4 rel0 4940 . . . 4  |-  Rel  (/)
5 wesn 4890 . . . 4  |-  ( Rel  (/)  ->  ( (/)  We  { (/)
}  <->  -.  (/) (/) (/) ) )
64, 5ax-mp 8 . . 3  |-  ( (/)  We 
{ (/) }  <->  -.  (/) (/) (/) )
73, 6mpbir 201 . 2  |-  (/)  We  { (/)
}
8 df1o2 6673 . . 3  |-  1o  =  { (/) }
9 weeq2 4513 . . 3  |-  ( 1o  =  { (/) }  ->  (
(/)  We  1o  <->  (/)  We  { (/)
} ) )
108, 9ax-mp 8 . 2  |-  ( (/)  We  1o  <->  (/)  We  { (/) } )
117, 10mpbir 201 1  |-  (/)  We  1o
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1717   (/)c0 3572   {csn 3758   <.cop 3761   class class class wbr 4154    We wwe 4482   Rel wrel 4824   1oc1o 6654
This theorem is referenced by:  psr1tos  16515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-suc 4529  df-xp 4825  df-rel 4826  df-1o 6661
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