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Theorem 0er 6711
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 4826 . . . 4  |-  Rel  (/)
21a1i 10 . . 3  |-  (  T. 
->  Rel  (/) )
3 df-br 4040 . . . . 5  |-  ( x
(/) y  <->  <. x ,  y >.  e.  (/) )
4 noel 3472 . . . . . 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 3472 . . . . . 6  |-  -.  x  e.  (/)
12 noel 3472 . . . . . 6  |-  -.  <. x ,  x >.  e.  (/)
1311, 122false 339 . . . . 5  |-  ( x  e.  (/)  <->  <. x ,  x >.  e.  (/) )
14 df-br 4040 . . . . 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 6702 . 2  |-  (  T. 
->  (/)  Er  (/) )
1817trud 1314 1  |-  (/)  Er  (/)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    T. wtru 1307    e. wcel 1696   (/)c0 3468   <.cop 3656   class class class wbr 4039   Rel wrel 4710    Er wer 6673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-er 6676
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