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Theorem cnveq0 5447
Description: A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
cnveq0  |-  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )

Proof of Theorem cnveq0
StepHypRef Expression
1 cnv0 5394 . 2  |-  `' (/)  =  (/)
2 rel0 5115 . . . . 5  |-  Rel  (/)
3 cnveqb 5446 . . . . 5  |-  ( ( Rel  A  /\  Rel  (/) )  ->  ( A  =  (/)  <->  `' A  =  `' (/) ) )
42, 3mpan2 669 . . . 4  |-  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  `' (/) ) )
5 eqeq2 2469 . . . . 5  |-  ( (/)  =  `' (/)  ->  ( `' A  =  (/)  <->  `' A  =  `' (/) ) )
65bibi2d 316 . . . 4  |-  ( (/)  =  `' (/)  ->  ( ( A  =  (/)  <->  `' A  =  (/) )  <->  ( A  =  (/)  <->  `' A  =  `' (/) ) ) )
74, 6syl5ibr 221 . . 3  |-  ( (/)  =  `' (/)  ->  ( Rel  A  ->  ( A  =  (/) 
<->  `' A  =  (/) ) ) )
87eqcoms 2466 . 2  |-  ( `' (/)  =  (/)  ->  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  (/) ) ) )
91, 8ax-mp 5 1  |-  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   (/)c0 3783   `'ccnv 4987   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996
This theorem is referenced by:  elrn3  29436
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