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Theorem 0er 6695
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 4810 . . . 4  |-  Rel  (/)
21a1i 10 . . 3  |-  (  T. 
->  Rel  (/) )
3 df-br 4024 . . . . 5  |-  ( x
(/) y  <->  <. x ,  y >.  e.  (/) )
4 noel 3459 . . . . . 6  |-  -.  <. x ,  y >.  e.  (/)
54pm2.21i 123 . . . . 5  |-  ( <.
x ,  y >.  e.  (/)  ->  y (/) x )
63, 5sylbi 187 . . . 4  |-  ( x
(/) y  ->  y (/) x )
76adantl 452 . . 3  |-  ( (  T.  /\  x (/) y )  ->  y (/) x )
84pm2.21i 123 . . . . 5  |-  ( <.
x ,  y >.  e.  (/)  ->  x (/) z )
93, 8sylbi 187 . . . 4  |-  ( x
(/) y  ->  x (/) z )
109ad2antrl 708 . . 3  |-  ( (  T.  /\  ( x
(/) y  /\  y (/) z ) )  ->  x (/) z )
11 noel 3459 . . . . . 6  |-  -.  x  e.  (/)
12 noel 3459 . . . . . 6  |-  -.  <. x ,  x >.  e.  (/)
1311, 122false 339 . . . . 5  |-  ( x  e.  (/)  <->  <. x ,  x >.  e.  (/) )
14 df-br 4024 . . . . 5  |-  ( x
(/) x  <->  <. x ,  x >.  e.  (/) )
1513, 14bitr4i 243 . . . 4  |-  ( x  e.  (/)  <->  x (/) x )
1615a1i 10 . . 3  |-  (  T. 
->  ( x  e.  (/)  <->  x (/) x ) )
172, 7, 10, 16iserd 6686 . 2  |-  (  T. 
->  (/)  Er  (/) )
1817trud 1314 1  |-  (/)  Er  (/)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    T. wtru 1307    e. wcel 1684   (/)c0 3455   <.cop 3643   class class class wbr 4023   Rel wrel 4694    Er wer 6657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-er 6660
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