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Theorem cnveq0 4777
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 4727 . 2  |-  `' (/)  =  (/)
2 rel0 4462 . . . . 5  |-  Rel  (/)
3 cnveqb 4776 . . . . 5  |-  ( ( Rel  A  /\  Rel  (/) )  ->  ( A  =  (/)  <->  `' A  =  `' (/) ) )
42, 3mpan2 401 . . . 4  |-  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  `' (/) ) )
5 eqeq2 2049 . . . . 5  |-  ( (/)  =  `' (/)  ->  ( `' A  =  (/)  <->  `' A  =  `' (/) ) )
65bibi2d 221 . . . 4  |-  ( (/)  =  `' (/)  ->  ( ( A  =  (/)  <->  `' A  =  (/) )  <->  ( A  =  (/)  <->  `' A  =  `' (/) ) ) )
74, 6syl5ibr 145 . . 3  |-  ( (/)  =  `' (/)  ->  ( Rel  A  ->  ( A  =  (/) 
<->  `' A  =  (/) ) ) )
87eqcoms 2043 . 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 98    = wceq 1243   (/)c0 3224   `'ccnv 4344   Rel wrel 4350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353
This theorem is referenced by: (None)
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