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Theorem 0er 7397
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 4969 . . . 4  |-  Rel  (/)
21a1i 11 . . 3  |-  ( T. 
->  Rel  (/) )
3 df-br 4418 . . . . 5  |-  ( x
(/) y  <->  <. x ,  y >.  e.  (/) )
4 noel 3762 . . . . . 6  |-  -.  <. x ,  y >.  e.  (/)
54pm2.21i 134 . . . . 5  |-  ( <.
x ,  y >.  e.  (/)  ->  y (/) x )
63, 5sylbi 198 . . . 4  |-  ( x
(/) y  ->  y (/) x )
76adantl 467 . . 3  |-  ( ( T.  /\  x (/) y )  ->  y (/) x )
84pm2.21i 134 . . . . 5  |-  ( <.
x ,  y >.  e.  (/)  ->  x (/) z )
93, 8sylbi 198 . . . 4  |-  ( x
(/) y  ->  x (/) z )
109ad2antrl 732 . . 3  |-  ( ( T.  /\  ( x
(/) y  /\  y (/) z ) )  ->  x (/) z )
11 noel 3762 . . . . . 6  |-  -.  x  e.  (/)
12 noel 3762 . . . . . 6  |-  -.  <. x ,  x >.  e.  (/)
1311, 122false 351 . . . . 5  |-  ( x  e.  (/)  <->  <. x ,  x >.  e.  (/) )
14 df-br 4418 . . . . 5  |-  ( x
(/) x  <->  <. x ,  x >.  e.  (/) )
1513, 14bitr4i 255 . . . 4  |-  ( x  e.  (/)  <->  x (/) x )
1615a1i 11 . . 3  |-  ( T. 
->  ( x  e.  (/)  <->  x (/) x ) )
172, 7, 10, 16iserd 7388 . 2  |-  ( T. 
->  (/)  Er  (/) )
1817trud 1446 1  |-  (/)  Er  (/)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   T. wtru 1438    e. wcel 1867   (/)c0 3758   <.cop 3999   class class class wbr 4417   Rel wrel 4850    Er wer 7359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-er 7362
This theorem is referenced by: (None)
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