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Theorem 0er 7347
Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
Assertion
Ref Expression
0er  |-  (/)  Er  (/)

Proof of Theorem 0er
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rel0 5127 . . . 4  |-  Rel  (/)
21a1i 11 . . 3  |-  ( T. 
->  Rel  (/) )
3 df-br 4448 . . . . 5  |-  ( x
(/) y  <->  <. x ,  y >.  e.  (/) )
4 noel 3789 . . . . . 6  |-  -.  <. x ,  y >.  e.  (/)
54pm2.21i 131 . . . . 5  |-  ( <.
x ,  y >.  e.  (/)  ->  y (/) x )
63, 5sylbi 195 . . . 4  |-  ( x
(/) y  ->  y (/) x )
76adantl 466 . . 3  |-  ( ( T.  /\  x (/) y )  ->  y (/) x )
84pm2.21i 131 . . . . 5  |-  ( <.
x ,  y >.  e.  (/)  ->  x (/) z )
93, 8sylbi 195 . . . 4  |-  ( x
(/) y  ->  x (/) z )
109ad2antrl 727 . . 3  |-  ( ( T.  /\  ( x
(/) y  /\  y (/) z ) )  ->  x (/) z )
11 noel 3789 . . . . . 6  |-  -.  x  e.  (/)
12 noel 3789 . . . . . 6  |-  -.  <. x ,  x >.  e.  (/)
1311, 122false 350 . . . . 5  |-  ( x  e.  (/)  <->  <. x ,  x >.  e.  (/) )
14 df-br 4448 . . . . 5  |-  ( x
(/) x  <->  <. x ,  x >.  e.  (/) )
1513, 14bitr4i 252 . . . 4  |-  ( x  e.  (/)  <->  x (/) x )
1615a1i 11 . . 3  |-  ( T. 
->  ( x  e.  (/)  <->  x (/) x ) )
172, 7, 10, 16iserd 7338 . 2  |-  ( T. 
->  (/)  Er  (/) )
1817trud 1388 1  |-  (/)  Er  (/)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   T. wtru 1380    e. wcel 1767   (/)c0 3785   <.cop 4033   class class class wbr 4447   Rel wrel 5004    Er wer 7309
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-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  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-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-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-er 7312
This theorem is referenced by: (None)
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