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Theorem 0er 6942
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 5001 . . . 4  |-  Rel  (/)
21a1i 11 . . 3  |-  (  T. 
->  Rel  (/) )
3 df-br 4215 . . . . 5  |-  ( x
(/) y  <->  <. x ,  y >.  e.  (/) )
4 noel 3634 . . . . . 6  |-  -.  <. x ,  y >.  e.  (/)
54pm2.21i 126 . . . . 5  |-  ( <.
x ,  y >.  e.  (/)  ->  y (/) x )
63, 5sylbi 189 . . . 4  |-  ( x
(/) y  ->  y (/) x )
76adantl 454 . . 3  |-  ( (  T.  /\  x (/) y )  ->  y (/) x )
84pm2.21i 126 . . . . 5  |-  ( <.
x ,  y >.  e.  (/)  ->  x (/) z )
93, 8sylbi 189 . . . 4  |-  ( x
(/) y  ->  x (/) z )
109ad2antrl 710 . . 3  |-  ( (  T.  /\  ( x
(/) y  /\  y (/) z ) )  ->  x (/) z )
11 noel 3634 . . . . . 6  |-  -.  x  e.  (/)
12 noel 3634 . . . . . 6  |-  -.  <. x ,  x >.  e.  (/)
1311, 122false 341 . . . . 5  |-  ( x  e.  (/)  <->  <. x ,  x >.  e.  (/) )
14 df-br 4215 . . . . 5  |-  ( x
(/) x  <->  <. x ,  x >.  e.  (/) )
1513, 14bitr4i 245 . . . 4  |-  ( x  e.  (/)  <->  x (/) x )
1615a1i 11 . . 3  |-  (  T. 
->  ( x  e.  (/)  <->  x (/) x ) )
172, 7, 10, 16iserd 6933 . 2  |-  (  T. 
->  (/)  Er  (/) )
1817trud 1333 1  |-  (/)  Er  (/)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    T. wtru 1326    e. wcel 1726   (/)c0 3630   <.cop 3819   class class class wbr 4214   Rel wrel 4885    Er wer 6904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-er 6907
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