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Theorem 0we1 6547
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 3493 . . . 4  |-  -.  <. (/)
,  (/) >.  e.  (/)
2 df-br 4061 . . . 4  |-  ( (/) (/) (/) 
<-> 
<. (/) ,  (/) >.  e.  (/) )
31, 2mtbir 290 . . 3  |-  -.  (/) (/) (/)
4 rel0 4847 . . . 4  |-  Rel  (/)
5 wesn 4798 . . . 4  |-  ( Rel  (/)  ->  ( (/)  We  { (/)
}  <->  -.  (/) (/) (/) ) )
64, 5ax-mp 8 . . 3  |-  ( (/)  We 
{ (/) }  <->  -.  (/) (/) (/) )
73, 6mpbir 200 . 2  |-  (/)  We  { (/)
}
8 df1o2 6533 . . 3  |-  1o  =  { (/) }
9 weeq2 4419 . . 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 1633    e. wcel 1701   (/)c0 3489   {csn 3674   <.cop 3677   class class class wbr 4060    We wwe 4388   Rel wrel 4731   1oc1o 6514
This theorem is referenced by:  psr1tos  16317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
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 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-suc 4435  df-xp 4732  df-rel 4733  df-1o 6521
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