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Theorem 0we1 6742
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 3624 . . . 4  |-  -.  <. (/)
,  (/) >.  e.  (/)
2 df-br 4205 . . . 4  |-  ( (/) (/) (/) 
<-> 
<. (/) ,  (/) >.  e.  (/) )
31, 2mtbir 291 . . 3  |-  -.  (/) (/) (/)
4 rel0 4991 . . . 4  |-  Rel  (/)
5 wesn 4941 . . . 4  |-  ( Rel  (/)  ->  ( (/)  We  { (/)
}  <->  -.  (/) (/) (/) ) )
64, 5ax-mp 8 . . 3  |-  ( (/)  We 
{ (/) }  <->  -.  (/) (/) (/) )
73, 6mpbir 201 . 2  |-  (/)  We  { (/)
}
8 df1o2 6728 . . 3  |-  1o  =  { (/) }
9 weeq2 4563 . . 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 1652    e. wcel 1725   (/)c0 3620   {csn 3806   <.cop 3809   class class class wbr 4204    We wwe 4532   Rel wrel 4875   1oc1o 6709
This theorem is referenced by:  psr1tos  16577
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-suc 4579  df-xp 4876  df-rel 4877  df-1o 6716
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