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Theorem cnveq0 4828
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 4778 . 2  |-  `' (/)  =  (/)
2 rel0 4511 . . . . 5  |-  Rel  (/)
3 cnveqb 4827 . . . . 5  |-  ( ( Rel  A  /\  Rel  (/) )  ->  ( A  =  (/)  <->  `' A  =  `' (/) ) )
42, 3mpan2 416 . . . 4  |-  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  `' (/) ) )
5 eqeq2 2092 . . . . 5  |-  ( (/)  =  `' (/)  ->  ( `' A  =  (/)  <->  `' A  =  `' (/) ) )
65bibi2d 230 . . . 4  |-  ( (/)  =  `' (/)  ->  ( ( A  =  (/)  <->  `' A  =  (/) )  <->  ( A  =  (/)  <->  `' A  =  `' (/) ) ) )
74, 6syl5ibr 154 . . 3  |-  ( (/)  =  `' (/)  ->  ( Rel  A  ->  ( A  =  (/) 
<->  `' A  =  (/) ) ) )
87eqcoms 2086 . 2  |-  ( `' (/)  =  (/)  ->  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  (/) ) ) )
91, 8ax-mp 7 1  |-  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   (/)c0 3268   `'ccnv 4391   Rel wrel 4397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-rel 4399  df-cnv 4400
This theorem is referenced by: (None)
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