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Theorem 0we1 7157
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 3789 . . . 4  |-  -.  <. (/)
,  (/) >.  e.  (/)
2 df-br 4448 . . . 4  |-  ( (/) (/) (/) 
<-> 
<. (/) ,  (/) >.  e.  (/) )
31, 2mtbir 299 . . 3  |-  -.  (/) (/) (/)
4 rel0 5127 . . . 4  |-  Rel  (/)
5 wesn 5071 . . . 4  |-  ( Rel  (/)  ->  ( (/)  We  { (/)
}  <->  -.  (/) (/) (/) ) )
64, 5ax-mp 5 . . 3  |-  ( (/)  We 
{ (/) }  <->  -.  (/) (/) (/) )
73, 6mpbir 209 . 2  |-  (/)  We  { (/)
}
8 df1o2 7143 . . 3  |-  1o  =  { (/) }
9 weeq2 4868 . . 3  |-  ( 1o  =  { (/) }  ->  (
(/)  We  1o  <->  (/)  We  { (/)
} ) )
108, 9ax-mp 5 . 2  |-  ( (/)  We  1o  <->  (/)  We  { (/) } )
117, 10mpbir 209 1  |-  (/)  We  1o
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1379    e. wcel 1767   (/)c0 3785   {csn 4027   <.cop 4033   class class class wbr 4447    We wwe 4837   Rel wrel 5004   1oc1o 7124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-suc 4884  df-xp 5005  df-rel 5006  df-1o 7131
This theorem is referenced by:  psr1tos  18039
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